measurement background booklet mb: teaching ......booklet mb: teaching measurement processes in...

YUMI DEADLY CENTRE School of Curriculum

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Measurement Background

Booklet MB: Teaching Measurement Processes in School Years P-10

YuMi Deadly Maths Past Project Resource

Acknowledgement

We acknowledge the traditional owners and custodians of the lands in which the mathematics ideas for this resource were developed, refined and presented in professional development sessions.

YuMi Deadly Centre

The YuMi Deadly Centre is a Research Centre within the Faculty of Education at Queensland University of Technology which aims to improve the mathematics learning, employment and life chances of Aboriginal and Torres Strait Islander and low socio-economic status students at early childhood, primary and secondary levels, in vocational education and training courses, and through a focus on community within schools and neighbourhoods. It grew out of a group that, at the time of this booklet, was called “Deadly Maths”.

“YuMi” is a Torres Strait Islander word meaning “you and me” but is used here with permission from the Torres Strait Islanders’ Regional Education Council to mean working together as a community for the betterment of education for all. “Deadly” is an Aboriginal word used widely across Australia to mean smart in terms of being the best one can be in learning and life.

YuMi Deadly Centre’s motif was developed by Blacklines to depict learning, empowerment, and growth within country/community. The three key elements are the individual (represented by the inner seed), the community (represented by the leaf), and the journey/pathway of learning (represented by the curved line which winds around and up through the leaf). As such, the motif illustrates the YuMi Deadly Centre’s vision: Growing community through education.

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© QUT YuMi Deadly Centre 2007 Electronic edition 2013

School of Curriculum QUT Faculty of Education

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Phone: +61 7 3138 0035 Fax: + 61 7 3138 3985

Email: [email protected] Website: http://ydc.qut.edu.au

CRICOS No. 00213J

This booklet was developed as part of a project which ran during 2007 and was funded by Australian School Innovation in Science, Technology and Mathematics (ASISTM): Using Finance and measurement applications to improve number understandings for Indigenous students. This was a joint initiative with Catholic Education and the Association of Independent Schools Queensland.

http://www.ydc.qut.edu.au/

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http://ydc.qut.edu.au/

ASISTM Cherbourg-Woorabinda Project: Using Finance and Measurement Applications to Improve

Number Understandings for Indigenous Students

DEADLY MATHS PROJECT

Measurement Background Booklet

TEACHING MEASUREMENT PROCESSES IN SCHOOL YEARS P-10

16/09/2007

Text and drawings prepared by:

Tom J Cooper

Annette R Baturo

Typed and drawings electronically prepared by:

Edlyn Grant

Imogene Grant

Deadly Maths Projects

Faculty of Education

QUT, Kelvin Grove, Qld, 4059

YuMi Deadly Maths Past Project Resource © 2007, 2013 QUT YuMi Deadly Centre

ASISTM Booklet MB: Measurement Background, 16/09/2007 Page iii

CONTENTS PAGE

PREFACE AND ACKNOWLEDGMENTS ........................................................................................... v

SECTION 1: OVERVIEW OF MEASUREMENT YEARS P-10 .................................. 1

OVERVIEW OF MEASUREMENT YEARS P-10 ................................................................................ 2

SECTION 2: THE FIVE STAGES IN TEACHING MEASUREMENT ....................... 12

CHAPTER 2.1: IDENTIFYING ATTRIBUTES FOR MEASURE ...................................................... 13

CHAPTER 2.2: COMPARING AND ORDERING ............................................................................. 20

CHAPTER 2.3: NON-STANDARD UNITS ........................................................................................ 28

CHAPTER 2.4: STANDARD UNITS ................................................................................................. 45

CHAPTER 2.5: FORMULAE ............................................................................................................. 62

SECTION 3: SEQUENCING THE TEACHING OF MEASUREMENT ..................... 71

CHAPTER 3.1: SEQUENCING MEASUREMENT ACTIVITIES ...................................................... 72

CHAPTER 3.2: TEACHING MEASUREMENT ................................................................................. 93


ASISTM Booklet MB: Measurement Background, 16/09/2007 Page v

PREFACE AND ACKNOWLEDGMENTS

In 1988, a Measurement booklet was prepared by Tom Cooper covering the teaching of

measurement through the primary school and into junior secondary school. It was part of a series of

booklets produced at that time by the lecturers of Carseldine campus of Brisbane College of

Advanced Education.

This booklet has been rewritten by Annette Baturo as part of a program to rewrite these booklets. This

early draft in this rewriting process has been put together to act as a background booklet to the

teaching of measurement for an ASISTM project titled, Using Finance and Measurement Applications

to Improve Number Understandings for Indigenous Students. The drawings have been electronically

prepared and placed by Edlyn and Imogene Grant.

The original 1988 booklet owed much to the MEASUREMENT book developed by Donald R Kerr Jr.,

Kathleen M Hart and Calvin J Irons for preservice teacher education students at Indiana University for

the Mathematics Methods Program (MMP) under the direction of John F LeBlanc and Donald R Kerr

Jr. (The preparation of the MMP books was supported by the National Science Foundation Grant GY-

9293. Copyright ended on the MMP materials on December 31, 1981.)

The booklet is in three sections. Section 1 overviews measurement and what is in the booklet. Section

2 outlines a 5 step process for teaching measurement with Chapters 2.1 to 2.5 as one chapter on

each step. Section 3 takes each attribute in turn and summarises how the 5 steps affect them

(Chapter 3.1) and looks at some teaching implications (Chapter 3.2).

Tom Cooper, 16/09/07


ASISTM Booklet MB: Measurement Background, 16/09/2007 Page 1

SECTION 1

OVERVIEW OF MEASUREMENT YEARS P-10

This small section overviews the booklet, covering relationships to other mathematics, the

5 stages for developing measurement, topics in measurement (length, area, volume,

mass, time, temperature, angle and value), and approaches to teaching measurement.



OVERVIEW OF MEASUREMENT YEARS P-10

BASIC MEASUREMENT IN P-10 MATHEMATICS

This book focuses on the basics of Measurement. The purpose of the book is, therefore, on how to develop

students’ understanding of the concepts of length, area, volume, mass, time, money, temperature and angle,

the units used, the formulae applied, and competence in the process of measuring, in using instruments and

in choosing appropriate units.

Measurement is an important human activity. It is an everyday skill. It is an essential tool of science, and it

provides a useful link between the real world and mathematics. Measurements are as diverse as the length

of a straight line, the IQ of a human being, and the speed of light. Measurement skills include simple

dexterity, the techniques of calculus, and the ability to construct models of human thought and behaviour.

This book concentrates on those aspects of measurement which are basic in the elementary school.

In simple terms, measurement involves the comparison of an amount of an attribute with a unit amount of the

attribute on order to determine a number. For example, to measure the amount of the attribute length

possessed by a pencil one might compare the pencil with a unit of length such as the centimetre to

determine a number, say 15.

Within the unique and important place mathematics has had in Western Civilisation, measurement plays an

important part. The two well springs of mathematical thought have been number and geometry and these are

also the starting points for primary mathematics. But the impetus for the development of number and

geometry has often been problems of measurement (e.g., astronomy, surveying, taxes) and measurement

still plays a crucial role in the opportunities it offers for consolidation, application and development of number

and geometric concepts and processes. It is convenient to consider the primary mathematics syllabus as

three major topic areas as is diagrammatically described in figure 1 below.

Figure 1. The relationship between Number, Geometry, and Measurement.

It is also convenient, although somewhat simplistic, to think of measurement as the application of number to

geometry (to shape and position). This book focuses only on the basic applications – length, area, volume

and capacity, mass, time, money, temperature and angle. Probability, statistics, graphs and charts, is the

focus of another book. Measurements such as IQ, velocity, force, etc., not being in primary school

mathematics, are not covered.

SEQUENCING MEASUREMENT ACTIVITIES

Measurement raises some problems for children. It requires the recognition of attributes which may not yet

have become conscious parts of a child’s experience. This frequently involves the introduction of new words

in the child’s vocabulary. The comparisons involved in measurement depend on a clear understanding of

what can be done to an object without changing the quantity of an attribute that is present. For example, is

MEASUREMENT

Linear, Area, Volume (liquid &

solid),Mass

Time, Money

Probability

Statistics, Graphs & Charts

GEOMETRY

Shape (1-D, 2-D, 3-D)

symmetry, congruence, similarity

tessellations

Location (coordinates)

Movement

NUMBER

Numeration

Operations

Fractions (decimal & common)

Percents, rate & ration

Early algebra



the volume of a quantity of liquid changed when its container is changed? Dexterity is also required to

manipulate measurement instruments. Consequently, children seem to require both experience and maturity

to grasp measurement skills and concepts. This need suggests a spiral approach to measurement

instruction, in which difficult concepts are visited and revisited throughout a child’s schooling.

Children need to come to terms with units, how they are used, the comparisons that can be made with them

and how they interrelate with the number that results. Children must become proficient with metrics and in

applying commonly used measurement formulae. For these reasons, it is recommended that all topic areas

in measurement be developed with children through the five stages below.

Identifying the attribute to be measured

Before they can compare or measure an attribute such as mass, children must be aware of what mass is.

This has to come from experiencing instances of the attribute. It requires careful development of language.

Much of this is completed by the time children reach school age but it is essential for teachers to check that

this has happened. For instance, many children mistake mass for size or volume, believing always that the

largest object has more mass.

One method of exhibiting a new attribute to a child lacking experience with it is to show the child:

instances where everything varies except the attribute (e.g., ribbons, sticks, cylinders, pens all with the same

length); and

instances where the only thing that varies if the attribute (e.g., pink ribbons of different length but of identical

colour, width and material).

Comparing and ordering the attribute (no numbers)

Before children use numbers in measuring situations, it is useful to experience the concepts of length, area,

etc, by comparing and ordering different examples. Children should undertake comparing activities between

two examples, before ordering three or more examples. The change from comparing two to ordering three or

more is difficult. It requires a focus on betweenness, in identifying the example which is between the other

two.

Comparing activities should include all the types below and should be introduced to children in the sequence

below:

direct comparison where only the attribute being compared varies (e.g., comparing 2 pencils of different

lengths);

direct comparison where more than the compared attribute varies (e.g., comparing the lengths of a pencil and a

pair of scissors;

indirect comparison via an intermediary (e.g., comparing the distance around a can to the length of a pencil);

comparing different representations of the attribute (e.g., comparing the diameter of a bicycle wheel with the

width of a door).

Ordering activities should include all the types below and should be introduced to children in the sequence

shown:

copying an ordered sequence of examples;

placing an example correctly in an ordered sequence of other examples when the example to be placed is such

that, in terms of the attribute, it is greater or lesser that all the other examples (i.e. it has to be places at an end

of the already existing sequence);

placing an example correctly in an ordered sequence of other anywhere; and

ordering three or more examples without any assistance.



Non-standard (or child-chosen) units

This is the first point at which number is introduced in the measurement activities. It is introduced with units

from the everyday world of the child not with metrics. For example:

handspans, paces, blackboard dusters can be used to measure length;

cups full, glasses full to measure volume; and

paper clips, books to measure area.

The reasons for using non-standard units are threefold:

they are more natural, personal and familiar (and more fun), are commonly used in real life and do not have the

added problem of notation and conversion factors;

they show that standard units are not absolute (but were chosen for reasons of history and convenience); and

they provide a vehicle for teaching the role of units in the process of measuring.

The objectives with regard to measuring that can be achieved by using non-standard units are:

when measuring, the unit being used must not change;

when comparing or ordering examples by number of units, the same units must be used in all measurements;

when using common units, the example with the largest number has the greatest amount of attribute and vice

versa;

in arriving at a number, the larger the size of the unit the smaller will be the number and vice versa (the inverse

relation property);

in assessing measurements, the numbers should not be seen as correct in the sense that 2 + 3 gives 5 but

should be seen as accurate to the nearest unit (accuracy vs correctness) – there should be a tolerance for error;

in measurement situations and with a variety of units, being able to accurately estimate the amount of attribute

in terms of those units;

in measurement situations, being able to choose appropriate units (matching unit size to the object being

measured and to the need for accuracy); and

in measurement situations, there is a need to standardise units.

Standard units

With the successful teaching of the need for standardised units comes the opportunity for instruction in our

society’s standard units – metrics. Primary schools use only some metric units as laid down in the System

Internationale conventions and these are:

(i) Length km = 1000 m (kilometre)

m = 100 cm (metre)

cm = 10 mm (centimetre and millimetre)

(ii) Area km2 = 100 h (square kilometre)

h = 10 000 m2

(hectare)

m2 = 10 000 cm

2 (square metre, square centimetre)

(iii) Volume and Capacity m3 = 1 000 000 cm

3 (cubic metre and square centimetre)

kL = 1000 L (= m3) (kilolitre)

L = 1000 mL (mL = cm3) (litre and millilitre)

(iv) Mass t = 1000 kg (tonne)

kg = 1000 g (kilogram and gram)

(1 g = mass of 1 cm3 of water at 4ºC; 1kg = mass of 1 L of water at 4ºC)

(v) Time century = 100 years decade = 10 years

year = 52 weeks = 12 months = 365 days



week = 7 days fortnight = 2 weeks = 14 days

day = 24 hours (h)

h = 60 minutes (min) min = 60 seconds (s)

(vi) Temperature freezing point of water = 0ºC (zero degrees Celsius)

boiling point of water = 100ºC

(vii) Angle 360º = full turn

Metrics (or any standard unit) should be introduced in three stages:

identification (constructing actual units out of materials, or experiencing examples of them);

internalisation (finding examples in students’ own bodies, or in common everyday items, that equal units in

size); and

estimation (being able to “think” in the units and visualise them sufficiently to be able to make good “educated”

guesses at the measure of examples).

The introduction of metrics is also the opportunity for continuing to develop the seven objectives discussed in

non-standard units.

Formulae

The final stage in the development of skill and understanding in measurement is the application of formulae.

It is a sad indictment of must of modern teaching of measurement that formulae is often the first instance

with which many children are introduced to measurement concepts and processes. For example, many

children have their first real experiences with area with the formula for the area of a rectangle. This means

that they have great difficulty understanding area for irregular shapes.

Formulae enable us to calculate measures of regular shapes from their dimensions – a much faster process

than counting units. The formulae most appropriate for primary schools are:

Perimeter Rectangle

2(L + W)

Circle

2 R

Area Rectangle

L × W

Triangle ½ (L × W)

Parallelogram

L × W

Circle

R2

Volume Rectangular prism

L × W × H

Prism

Area of base × H

(½ (L × W) × H

Cylinder

Area of base × H

(R2 × H)

L

W

L

W

L

W

L

W

LW

H

LW

H

HH

H

R

H

R



Formulae can be directly applied to examples of the shape or may require analysis and synthesis as is

shown below:

(1)

What is the area of the concrete path?

Area of large rectangle = 6000 m2 (60 m × 100 m)

Area of small rectangle = 80 m× 40 m = 3200 m2

Area of path = 2800 m2

(2)

What is the area of the shape on the left?

Area of rectangle = 6000 m2

Area of triangle = ½ (30 m × 30 m) = 4500 m2

Area of semicircle = 1/2 ( 3.14 × 15 m × 15 m)

= 353.25 m2

MEASUREMENT TOPICS

There is a famous saying of Thorndike: Anything that exists, exists to some degree and can be measured.

Hence there is a need to develop a general understanding of the measurement process that could be

applied to any attribute. But, within primary school, there are particular attributes which have to be

specifically covered. These are discussed below.

Length

This is a one-dimensional concept related to the geometric notions of direction and line. It is a measure of

the separation, of two points along a straight line (a curve is measured by dividing it into small straight

segments). When related to plane shapes, particular lengths are perimeter/circumference, diameter, and

radius.

Area

This is a planar or two-dimensional concept related to the geometric notion of region. It is a measure of the

amount of coverage of that region formed by two lengths in two directions.

Capacity/volume

Capacity is a three-dimensional concept related to the geometric notion of a solid. It is a measure of the

amount of space enclosed by a solid shape. The interior of solid shapes can be filled or partially-filled with

liquid or solid materials (e.g., rice, soil, blocks). The volume of the substance can be measured. Put more

simply, capacity refers to the measurement of the entire enclosed space (e.g., “filled to capacity”) whilst

volume refers to the measurement of the amount of space taken up by a substance. When the substance

completely fills the solid shape, then the capacity and volume measurements are equal.

Mass

Although solid shapes have mass as a property, mass is not directly related to solid shape. Rather, it is

the measure of the inertia of an object, that is, how much force it takes to move the object, and how much

force it takes to stop the object from moving. This may not be proportional to the volume of solid shapes.

60

m

10 m

100 m

60

m

10 m

100 m

60m



For school students, it is related to the force exerted by an object as a result of gravity (the weight of the

object) – and can be intuitively understood as the amount of pressure the object exerts down on the hands

holding it up (the “heft” of the object). It should be remembered that weight varies depending on the planet

the person is standing on. (On the moon, our weight is approximately 1/6 of what it is on earth.) Mass does

not vary. Invariant mass has replaced varying weight as the “inertial” attribute of an object which is studied

by school students.

Time

Time is often called the fourth dimension. It is the duration we spend undertaking an activity. We are fixed

within the flow of time. We are unable to move through time at our own volition or at our own speed and

direction. Students are continually experiencing time. We do not have to set up special “changes” for them to

experience. They only need to be directed to reflect on their own experience. But, of course, we cannot set

up any contrasting experience (such as no change in time or twice as fast a change in time) with which to

compare. Time is also a very subjective experience.

Money/value

Money is the units for cost or value that we put on objects. As a measure, it is not as absolute as length but

varies according to demand. It may seem strange to place money within the measurement context but

measurement teaching approaches can be useful in instructing students about money.

Temperature

Temperature is a measure of how hot or cold objects are – a very subjective experience without measuring

instruments. The most important reference points are the freezing point and boiling point of water.

Angle

Angle is the amount of turn – a measure of change in direction. As a concept it is contained both within

geometry and measurement.



APPROACHES TO TEACHING MEASUREMENT

The topic of measurement contains both concepts and processes to be learnt. The processes consist of

general ones which pertain to measuring with units and particular ones which are related to rules and

formulae.

Activity types

The concepts and processes in geometry are best developed by the use of exemplars (materials or

pictures). Exemplars particular to geometry consist of measurement equipment such as

rules, tape-measures, trundle wheels

grid paper

measuring cylinders, blocks, containers

beam balances, spring balances, weights

clock faces, timers, eggtimers

play money, money stamps

thermometers

protractors, rotograms

Once the appropriate materials have been chosen, behaviour that may indicate that the content to be

covered has been learned must be identified, kept in mind during teaching and used for evaluation. Then

teaching activities that keep a balance between developing the ideas, consolidating, the ideas and applying

the ideas should be prepared and undertaken as Figure 2 below diagrammatically shows.

Figure 2. Wilson cycle

The approach of teaching should be to ensure that the children integrate the new ideas into their already

existing knowledge to form conceptual schemas – that the ideas are not retained as isolated pigeonholed

pieced of information but interconnected with previous ideas. To this end, schematising activities are

important – there should be a focus on, e.g., connecting the formulae for area of a triangle with area of a

rectangle so that both can be recalled from one memorisation.

Active teaching

Measurement concepts and processes (particularly formulae) should be discovered by children. Being

basically a procedural knowledge (a knowledge of how to do something), measurement is best learnt from

the children’s own experiences – by doing. Teaching should be active not imitative as Figure 3 shows.



Teach actively … … not imitatively

(1) Children explore (1) Teacher describes and explains the idea

(2) Children record and analyse results looking for pattern;

(2) Teacher demonstrates the ideas or the procedures that use the idea

(3) Children “discover” ideas/procedures; (3) Children internalise the idea/procedures;

(4) Children practice these ideas and the

procedures that depend on them.

(4) Children work similar examples using the

ideas/procedures.

Figure 3. Teaching actively/imitatively.

It is important that, in the development of ideas and procedures, exemplars (models), language and symbols

be interrelated. This means that during the development and consolidation of an idea that teaching activities

vary to include all six interactions.

Teacher asks questions using Children reply using

Exemplars/models Language

Language Exemplars/models

Exemplars/models Symbols

Symbols Exemplars/models

Language Symbols

Symbols Language

It is important that children experience: (i) many embodiments (examples using different materials or

showing different types) of concepts/processes; and (ii) both examples and non examples of

concepts/processes.

General objectives

There are many specific objectives within the area of mathematics that teaching has to meet – specific

activities in length, area, volume, etc. But it is important to remember that while these are being achieved,

there are higher level more general objectives that should also be the aim of instruction. Some of these are

described over.

Estimation

Thinking in terms of measurement requires the ability to estimate in terms of units. Children should be given

many opportunities to estimate lengths, areas, masses, etc. In fact, it is a good idea to require an estimate

before any measure is made.

Reasoning

It is important that thinking skills be developed in measurement situations. Many practical measurement

activities and applications of formulae require children to solve problems. Such activities should be included

to ensure that children can reason with measures.



Applications

Measurement offers great opportunities for real world applications (e.g., the cost of carpeting the school).

This should be a major focus of measurement teaching.

Appreciation of the use of measurement

Measurements makeup a large part of the information we use to make judgements. Are these

measurements appropriate and valid for the questions we face? Are measurements an appropriate source of

data for the decisions? Children need to grow to appreciate when measurement is useful (and when it is not)

and the extent that its use if valid.

Interrelating attributes

When objects are made of the same material, mass is proportional to volume. The volume of a prism relates

to height and the area of its base. In a similar way other attributes are related to each other. Children should

know when and how the various attributes are related.

Developing measures

In their study of length, area, etc, children should acquire the general ability to work with measures in

investigations. This includes

finding and defining a problem to be investigated;

designing the investigation and determining the measures to be undertaken;

designing the instruments to do the measurement with (where necessary);

determining the criteria for calibrating the instruments (where necessary).

Examples of questions to be investigated which are sufficiently open ended to allow wide scope to the

children are determining the best seat to sit in, in the classroom, determining the happiest person in the

school, etc.

Developmental level

Measurement is on of the areas of mathematics for which there is some evidence that age and maturation

place limits on what children can do. For this reason, teachers need to watch the following:

Motor skills

Children must coordinate reasonably complicated motor skills to undertake some measurements accurately.

For example, rulers must be placed in a straight line and meet end to end for length and water must be

poured accurately into cylinders for volume. Some children find it hard, for example, to align the end of an

object with the zero of a ruler when measuring length of not to be clumsy when measuring capacity by

immersion and overflow. Teachers should expect only the accuracy children are capable of.

Conservation

Young children may find it difficult, in some situations, to realise that lengths stay the same when one object

is placed above another, or that volume remains the same for an amount of water regardless of whether the

glass is short and wide or tall and narrow or that mass is the same for a ball of plasticine as it is when the

plasticine is rolled out into a “snake” etc. Such activities should be given to all children but correct answers

only expected from children whose development is such that they are capable of answering correctly.

Mass is a particular problem. Children commonly confuse mass with volume/size or with quantity. They think

larger size means greater mass and, egg five foam balls are heavier than one steel ball because there are

more of them.



History

Measurement has a colourful history. This can be used to enliven its teaching. Unfortunately the Metric

system is much younger than the old British system of units and has a much more clinical history still, time

and angle go back to Babylonia and the history of the Napoleon inspired “scientific” development of metrics

by the French can be contrasted to the more “natural” growth of the British system out of body

measurements. Children can then see that, although metrics are much more logical in their conversion rates,

the British units were a more understandable and manageable size. For example, inches are related to

finger-size, feet to foot size and yard to arm length. Metres and centimetres are not so easily related and not

such a nice manageable size.

Connections to other topic areas

Possibly the most important teaching approach is to connect the teaching of measurement to number and

geometry.

Length, area, volume and capacity, mass and angle can easily be related to shape (both line, plane shape and

solid shape) and measures such as length and angle can be used to classify shapes and are the basis of

concepts such as similarity and congruence.

Metrics and money can easily be related to decimals (in actuality, this is a relation which is essential for the

development of all three areas of metrics, money and decimals).

Measurement topics provide excellent opportunities to apply (and consolidate) number knowledge.

In terms of actual activities, it is an excellent idea to:

Teach metrics along with decimals – dollars and cents and metres and centimetres with 2 decimal places and

litres and millilitres and kilograms and grams with three decimal places; and

Continue work on solids and plane shapes into length (perimeter), area, volume and mass, and then use these

measurements to classify the shapes (egg isosceles triangles).

SUMMARY

The basis of this book lies in the sequence of measurement activities described above and diagrammatically

represented in Figure 4.

Figure 4.* Sequence of measurement activities.

*The sequence in Figure 4 above is a cornerstone of this book, but it should not be seen as a hard and fast rule, particularly in terms of

finishing one stage before starting the next. It is very possible, and even commendable, to start, say, standard units while still

completing non-standard work.



SECTION 2

THE FIVE STAGES IN TEACHING MEASUREMENT

This section is 5 chapters to match the 5 stages. Chapter 2.1 looks at identifying the

different attributes and recognising differences and similarities (or relationships) between

different attributes. Chapter 2.2 focuses on comparing and ordering different amounts of

attributes without using units and number. Chapter 2.3 introduces units and number into

measurement via non-standard or child-chosen units. Chapter 2.4 focuses on standard

units – metrics, hours, minutes, etc dollars and cents and degrees (for both angle and

temperature). Finally, Chapter 2.5 looks at formulae and their application in measurement

situations.



CHAPTER 2.1: IDENTIFYING ATTRIBUTES FOR MEASURE

Being able to identify the attribute that has to be measured is obviously the first step in developing skill and

understanding in measuring that attribute. This chapter focuses on how this is to be done.

Children come to school with some ideas of the major attributes. Length is usually reasonably well known, as

is volume, but mass is often misunderstood and area may not be know at all. There is some understanding

of money, time and temperature but angle usually needs formal development. The early teachers job is to

check what is know, build what is not known and flesh out what is partially know.

But identifying attributes is not something that should be left only to Grade 1 teachers. Many children take

some years and many experiences to gain complete schematic understanding of notions such as area,

volume, mass, money or value, time and temperature. Witness the difficulties most people have with the

change to metrics – many people rely on a symbolic rote understanding of attributes and do not have a

meaningful understanding that transcends unit names and sizes.

Furthermore, there is a need to develop in children a general knowledge of how to identify attributes whose

measurement would help them to solve problems. This requires the ability to define and categorise, to

construct measuring instruments and to calibrate them. And much of the later formulae work relies on

relationships between different attributes – an area that builds on individual attribute understandings. Hence,

Unit 2.1.1 will focus on identifying attributes, Unit 2.1.2 will look at relationships between attributes and Unit

2.1.3 will look at instrumentation – how to develop, calibrate and use instruments appropriately for the

attribute under investigation.

Unit 2.1.1: Identifying attributes

Focus

In this unit we focus on enabling children to identify the attributes of length, area, volume, mass, time,

money, temperature and angle.

Background

If you do not know what it is, you cannot measure it. The first task of measurement instruction is to ensure

children are aware of what it is we want them to measure. (Read carefully the discussions of the attributes in

the overview of this book.) This awareness must extend to all the words that are associated with that

attribute. This can be quite dramatic a list as the words for length below give some indication:

long longer longest short shorter shortest

wide wider widest narrow narrower narrowest

high low up down

above below near far

thick thicker thickest thin thinner thinnest

close away next to distant

tall taller tallest

height width shortness narrowness

thickness thinness distance



Activities

Materials Required: Scissors, tape, paper, cardboard, pens, pencils

Acting in groups, look amongst the pens, pencils scissors and other materials in the room and find as many different

articles as you can that are the same length. Cut pieces of cardboard and paper (of different colours, widths and

thicknesses), ribbon, string, tape, etc, to equal this length. Call this collection Set A.

Acting in groups, cut strips of paper of equal width (and colour) but differing lengths. Call this collection Set B.

Spread Set A and Set B separately on your table. Consider that you have a young child who is looking at the two sets.

The following questions could direct children to the attribute length.

“What is the same about the objects in each set?”

“What is different about the objects in each set?”

For which set (A or B) would you expect the questions to cause the most difficulty?

What could we do with the items in Set B to make the attribute of length more obvious to children?

In the above activity, Set A is composed of items of equal length where other attributes vary and Set B is composed of

items of differing length which are otherwise the same. What collections of items could you make to achieve the

same purpose for:

Area?

Volume?

Mass?

Time, Money (Value), Temperature and Angle require more thought in introducing to children.

Angle can be introduced by looking at things that turn (door knobs, wheels, drills etc). What could we

look at to introduce:

Time?

Money (or Value)?

Temperature?

To introduce children to temperature, a teacher visited a cold storage warehouse and the children were allowed to stay in

the refrigerated room for a short time. How could we use this experience to highlight temperature? What

questions would you ask?

What similar excursions/activities/experiences would assist children to focus correctly on:

Time?

Money (or Value)?

Mass?

Pasting paper on objects to cover them is an excellent introduction to area. What other activities could we do to introduce

area?

Temperature?

Volume?

Mass?



In the background to this unit we listed a collection of words which are used in reference to length. List a similar

collection for:

Volume?

Temperature?

Money or Value?

Acting in groups, secretly look at the members of your group to identify human attributes that all members have to a

lesser or greater degree. Choose one of these attributes (it may be colour of hair, length of fingernails or the

amount of blue being worn) and decide which member has most/least of it. As directed by your lecturer, present

yourselves to the rest of the class lined up so that the ordering from most to least (left to right) is evident. The rest

of the class must guess you attribute. Try to choose something that is fun, unusual and “clever” but not impossible

to guess.

From the attributes below, determine how you would describe each attribute to people who were not familiar with it,

describe some experiences you might give to these people to make them familiar it and describe procedures

which could be used to compare and order:

Colour

Intelligence

Pain

Teaching hints

Use the everyday experiences of the children. This is particularly valuable for attributes such as length, time, money

(value) and temperature.

Exploration and questioning should be the major tools of teaching. An understanding of an attribute cannot be told to a

child.

Materials must be experienced by children. Children should be given every opportunity to heft different objects, build with

bricks, pour water and sand, etc.

The teachers role should be to encourage language (appropriate to the attribute) and manipulative skills. The teacher

should also prepare the child for the next stage by drawing out ideas of relative size.

Use a variety of activities. For example, for length:

name tall short narrow……objects

tie string or ribbon together

unroll balls of string

draw short or long lines

sort objects by length

cut streamers as long as your pen

direct students to follow instructions (e.g., walk to the low tree)

make a tall tower of Lego

Teachers should remember that, in life, comparisons are continually made on the basis of many attributes

(e.g., beauty, temperatures, velocity and size). Some of these attributes (e.g., humidity, intelligence or

warmth) may be as nebulous to them as mass, for example, is to children.



Unit 2.1.2: Relationships between attributes

Focus

In this unit we explore the extent to which differing attributes e.g., mass and volume, are related to each

other.

Background

There are definite relationships between some attributes that show up in formula. Other attributes are not

generally related but from particular relationships in some special instances. Other attributes are

independent and form no relationships.

Some of the relationships that are worth exploring with children are below:

1. Volume, area and length – these are directly related as is shown in the formulae for area and volume.

2. Mass and volume – these are not generally related but are proportional when density remains the

same.

3. Time and temperature – these are independent of the other attributes and each other.

4. Money – the value of a particular substance is proportional to its size and mass but value (money) is

not generally related to mass or volume.

5. Angle and length – these are directly related as long as the length is long the arc a fixed distance from

the centre of the turn.

Activities

Materials required: Pen paper.

1. Young John believes the pillow is heavier than the brick. What is John focussing on? Mass or Volume? How can

we get him to see his error?

2. John believes that the large vase will be heavier then the small vase. Is this always true? When is it true? In what

situations could it be false?

3. A group of children were throwing objects of different masses and measuring how far they went. For a particular

child, how would you expect distance thrown to relate to mass? Would this always be so?

4. What would be the mass/length relationship if the objects were being hung from large rubber bands?

The children were playing a game – objects of different masses were being placed in a large ice cream container and the

lib put on and the children had to guess which object by hefting the container.

Why is this game so suitable for introducing mass?

Investigations

What would happen to the height of a litre milk container if we doubled the dimensions of its base?

Debate this position – “large things are more expensive than small”!

Why are cheap houses square? (Hint – look at relation of perimeter to area!)

Why do we have to be careful about babies getting too hot or too cold? (Hint – look at relation of volume/mass

to surface area!)

Why do large 1kg parcels heft lighter than small 1kg parcels?



Teaching hints

The world of merchandising and packaging is a constant source for interrelating attributes.

Why does a litre cold drink bottle seem so much larger than a MAB block?

How can a package be made look expensive? Or Cool?

Why does the 331/2 segment of the turnip look bigger than that of the circle?

Which contains the most?

Even as adults, we are constantly fooled by packaging. We do not have good concepts of, e.g., the attributes

of mass and volume.

Note: Developing an understanding of attributes should continue all through primary school.

Unit 2.1.3: Instrumentation

Focus

In this unit, we shall focus on instrumentation – how to develop, calibrate and use instruments appropriately

for the attribute under investigation.

Background

In many cases, the measurement of an attribute (e.g., intelligence) requires the development, calibration and

use of a special instrument. Refinements of measure (e.g., accuracy to thousandths of a millimetre) have

also required the development through technology, of special instruments.

It is important that children be made aware of the role of instrumentation and its relationship to the attribute

under investigation. It is particularly important that they be made aware of how an instrument that gives good

range of measures can come to change perception of an attribute and seduce the measurer into replacing

the attribute by the instrument. For example, IQ measurements on Stanford Binet tests replaced intelligence

(for many years) as the subject under study – intelligence became synonymous with this test. In the end, it

could be argued that researchers were not studying the effect of intelligence but the effect of test score

results from this instrument.

Activities

Materials required: Pen and paper; Materials for the investigation

1. Before we can develop instruments, it is necessary to understand what is involved in measurement in terms of

attribute, comparison and unit (so that an instrument can be developed which will correctly enable comparison

with a unit).



Example 1: Angle

(1) Describe in your own words the attribute being measured when you measure an angle. What

would be appropriate units for measurements? What would be an appropriate measuring

instrument for angle?

(2) Why would a child consider that the two angles represented below have different measures?

What “attribute” are children considering when they make such a mistake? What experiences

could you give such children to correct the misunderstanding?

(3) Look at a rotagram and a protractor. Why are these both good instruments for measuring

angle? Why is the degree, the radian, and the “whole turn” appropriate units?

Example 2: Temperature

(4) How does a thermometer actually measure temperature? (What physical principles are

involved?) Why can this instrument use height to measure temperature? Draw and label the

parts of a thermometer.

Example 3: Probability

(5) What does probability measure? What is the relation between statistic and probability? If seven

heads result from ten flips of a coin, what are the implications for probability (which would have

predicted 5 out of 10 heads)? What kind of instrument could be used to measure probability?

2. There are differences of opinion between educators on the importance of measuring human ability.

Teachers, of course, do a lot of such measurement (e.g., tests, assignments, projects) and use the

results to make crucial decisions about children. For the instruments below, discuss:

(1) the attribute being measured,

(2) the conditions under which an individual or a group can possesses more of this attribute,

(3) the errors inherent in the instrument,

(4) procedures likely to give rise to errors, and

(5) the potential advantages and dangers in the instrument.

IQ tests

School or College grades

Attitude questionnaires

Vocational preference scales

Standardised Arithmetic tests

Gallup polls

What measures of human ability seem to be the least prone to misunderstanding and error? What human

characteristics are the easiest to measure?

3. Investigations

(1) Choose a question that requires the development of a measuring instrument, for example:

What’s the happiest time of day?

What are the best type of TV programs?

What is the most comfortable chair in the institution?

How much is a degree worth?

(2) Design and administer the instrument.



(3) Write up a report of your question, your instrument design and your findings.

(4) Discuss the validity of your instrument for the task it was used for.

Teaching hints

It is important to allow children freedom for these investigations. They should see it as something they are

controlling.

The role of the teacher is facilitator – to react the children’s difficulties/impasses. The teacher has to

encourage, suggest ideas or new directions and supply resources. It is not necessary to have the answers –

in many cases there are no correct answers but just a lot of options.

The reporting back is an important part of the investigation and is the point, which may require much teacher

intervention. To steal an idea from the process approach to writing, teachers could conference each group

with rough drafts of their presentations before the final one is completed.

Investigations can be an open-ended form of instruction. Once the questions to be investigated are raised,

how the lesson proceeds can be determined by the interest of the children. The questions can be the basis

for whole-class or small-group discussion and activity. Such activity need only proceed for a short time, or

the investigation can be expanded into a “full-blown” series of activities. The children can be encouraged to

pursue anything that is of interest to them – as far as they wish to go.



CHAPTER 2.2: COMPARING AND ORDERING

Many of the real world situations involving measurement do not use units. For example, working out whether

a table will fit through a door, sawing a plank to make a shelf, or cutting paper to go round a bucket. In these

situations we can use wither direct or indirect comparison to determine what to do. For example, we can use

out bodies to check the width of the table against the door, we can place the plank beside the cupboard and

mark its length direct and we can use a piece of string to transfer the bucket’s circumference to the paper.

There are many other situations where we order objects without using units. For example, in buying stepping

stones we may select the widest in a collection, in choosing cucumbers we may select the longest

displayed, in choosing what work to do on a family camp, we may select what will take the shortest time, etc.

For these everyday occurrences and because the activities extend their understanding of the various

attributes, children should experience activities in which they compare and order attributes. To this end, the

units in this chapter have been designed so that Unit 2.2.1 focuses on direct comparisons, Unit 2.2.2 extends

this to indirect comparisons and Unit 2.2.3 focuses on ordering amounts of attribute.

A good ability to compare and order depends on rich experiences and the ability to visualise amounts. It also

depends on an imaginative approach to the use of everyday items as intermediaries in the comparison

process. For example, you would know that a board was not long enough to go right along the side of your

garage if it was shorter than your car.

Unit 2.2.1: Direct comparison

Focus

This unit looks at methods for directly comparing attributes and how these can be related to appropriate

teaching experiences for children.

Background

Each of the various attributes that make up the Measurement area of primary mathematics has different

comparison techniques. For example:

length – requires two objects to be placed beside each other with one end aligned;

area – requires the objects to be placed on top of each other for overlap;

volume – sand or water can be poured from one object to the other if it is hollow, solid objects would have to be

immersed to see differences in water level rise;

mass – these can be directly compared on a beam balance but each arm length must be the same;

time – requires the two activities to be run concurrently (as a “race”), starting together;

money or value – this depends on personal preference unless units are used and this could only be determined

by what children were willing to do for the items;

temperature – by feel; and

angle – by placing a copy of one angle over top of the other.

Activities

Materials required: Wire coat hanger, string, margarine containers, large ice-cream containers, assorted

materials for measuring.

1. Construct a beam balance out of a wire coat hanger, string and two margarine containers. Use it to compare:

a duster and ten pieces of chalk;

a stapler and 3 pairs of scissors; and

other items as directed by your instructor.



2. Half fill a large ice-cream container with water. Mark the water height with a pen. Immersing objects will raise the

water level. Which of the following objects raises it the most:

a plate or ten Unifix cubes;

three MAB flats or a coke can; and

other items as directed by your instructor.

3. Here are some comparison activities.

A group of children held “races” amongst themselves comparing, for example, taking off and putting on their

shoes with running around the school.

Another group of children were discussing what clothes would be comfortable outside versus the air-conditioned

room.

Using these as examples, think up some of your own activities to compare

Area Value

Angle Length

4. John placed the two sticks as on right and said that the “fat” one was longer.

What is the problem here? What can be done about it? (Children do not seem to

have a problem in comparing heights – could this be used to help John?)

5. Frank placed the red table on top of the blue table as shown below but it did not help

him to see which covered the most area? What could he do?

6. The teacher stood on one end of the board as below and none of the children

could lift him off the ground when they stood on the other end. Then Jenny

said she knew how to do it. She shifted the brick and lifted the teacher.

Where did she shift the brick to?

What can this activity do to help children’s understanding of beam

balances?

7. Jack and Bill were seeing if writing their name 5 times was quicker than 20 hops. “Go” said Anne. But bill was still

searching for his pencil when Jack started hopping. How can we lead them to see the problem here?

Teaching hints

These activities will give some insight into whether children are fully familiar with the attributes being

compared. Teachers need to keep “one eye” open for this and use the activities to help consolidate these

attribute understandings.

Make the activities fun. Rely on discussion afterwards, where children describe what they have done and

found out, to draw out problems of aligning ends or sides or starting points.

Some attributes (e.g., temperature, value) are very subjective in comparison – this should be kept in mind

and discussion, not “correct answers”, should be the focus of the activity

We should never forget to connect activity to the real world. For example, when a farmer was asked which

property he would like (he was shown a picture of a large area of land and a smaller area of land), he could

not give an answer because he did not know the quality of the soil, the availability of water, etc.

Many decisions are not made on one attribute alone but on the interaction of many.



Unit 2.2.2: Indirect comparison

Focus

In this unit, we will extend comparing activities to those many situations where it cannot be directly achieved.

Background

In the real world, you often need to know whether, for example, the piano will fit before you move it to that

position. In these situations where you cannot, or do not want to, move one item alongside the other, an

intermediary has to be used. For the various attributes that make up primary measurement, examples of

intermediaries are:

length – string, parts of out bodies;

area – paper (cutting and pasting)

volume – amounts of water or sand;

mass – plasticine or sand or rice or etc;

time – shading a long rectangle on a blackboard;

money or value – gold;

temperature – amount of clothes; and

angle – rotagram or paper cut-out of the angle.

Activities

Materials required: String, scissors, glue, tape, paper, tangrams, rotagrams.

1. Use a piece of string to compare:

the width of the door to the height of the blackboard;

the height of the table to its width; and

any other items as directed by your instruction.

2. In Unit 2.2.1, Frank’s problem with the red and blue tables could be solved with paper – but what do we have to

use scissors to do?

3. Cutting and refitting are important techniques for developing area understanding. Tangram activities can be used

to teach that areas that are cut and reformed stay the same. Complete the tangram activities at the end of this

unit.

4. Rotagrams are an intermediary material for comparing angles. Use them to complete the angle activities at the

end of this unit.

5. Another example of the use of an intermediary is using string to measure a length of hoop iron to go around an

old cart wheel. Think up situations where intermediaries could be used in situations where we are comparing:

(1) Length (2) Area

(3) Volume (4) Mass

(5) Time (6) Money or Value

(7) Temperature (8) Angle

Teaching hints

In developing young students’ comparison techniques, we should move through four stages as below:

Direct comparison where only the attribute being compared varies (e.g., comparing 2 pencils of different length).



Direct comparison where more than the compared attribute varies (e.g., comparing the lengths of a pencil and a

pair of scissors).

Indirect comparison via an intermediary (e.g., comparing the distance around a can to the length of a pencil).

Comparing different representations of an attribute (e.g., comparing the diameter of a bicycle wheel with the width

of a door.).

Teachers should make every endeavour to relate these comparison activities to real world situations – to make them real

life.

Unit 2.2.3: Ordering

Focus

In this unit, we look at the change from comparing two examples in terms of amount of attribute to order

more than two examples.

Background

Understanding the process of ordering items by, for example, length or area is not a simple extension of

comparing. It relies, at its basis, on an understanding of transitivity – that, for example, is A is longer than B

and B is longer than C, then A is longer than C. Transitivity is a concept that is associated with maturation

and development and may not be available to some children.

For this reason, the following sequence of activities is recommended.

Copying an ordered sequence of examples.

Placing an example correctly at the end of an ordered sequence to complete it.

Placing an example correctly in an ordered sequence to complete it.

Ordering three or more examples without any assistance.

Activities

Materials required: Cardboard for the Tall Men’s Task and materials for chosen attribute in Activity 2.

1. Read the Tall Men’s Task (prepared by Dr Rod Nason) below. Make up the cardboard lengths and trial

the activities with another member of your group.

Tall Men’s Task

Material: Six pieces of cardboard each of (noticeably) different length and colour. (In order of length these are:

pink, white, yellow, blue, red, green.)



Teacher Action Child Response

Subtask A

Place cards randomly on table

Say: Show me the red card.

Say: Show me the green card.

* For children unable to complete this task, the teacher must select all future colour cards for the child.

Place green and red cards next to each other with bases aligned in this manner:

Say: Which card is longer?

Say: Which card is shorter?

Subtask B

Say: Show me the blue card?

Say: If you are going from the shortest to the longest (teacher points and moves hand from right to left), where should you place the blue card?

(Have child predict and then place card.)

If the child places the card this way: blue

Say: Is the blue shorter than the green and red?

If the child says yes, move blue card to correct position and say: Is the blue now shorter than the green and red?

* This is the end of task 1 for children unable to complete this subtask.

Subtask C

Say: Show me the yellow card?

Say: If I am going from the shortest to the longest (teacher points), where should you place the pink card?

(Have child predict and then place the pink card.)


Subtask D

Say: Show me the white card?

Say: If you are going from the shortest to the longest (teacher points), where should you place the white card?

(Have child predict and then place the white card.)

If incorrect, give back white card, remove blue, red and green and try again to get child to correctly place the white card (when only pink and yellow are present).

If still incorrect say: Is the white longer than the pink?

Say: Is the white shorter than the yellow?

Say: Are they going from shortest to longest?

(If child sees error, ask child to place correctly).


Subtask E

Shuffle cards and say: Put these cards in order from shortest to longest. (If child has difficulty, ask to locate smallest card, next smallest and so on).

If order completed, say: Point to the longest card.

Say: Point to the shortest card.

Say: Point to the second longest card.

Say: Point to the second shortest card.

Say: Point to the third longest card.

2. Choose one of the attributes below. Prepare materials, a series of activities and questions to lead

children through the four steps in the sequence for introducing ordering for this attribute.

(1) Area (2) Volume



(3) Mass (4) Time

(5) Temperature (6) Angle

Teaching hints

The teaching of the ability to order items by attributes must be extended to non-standard and standard units.

The work here should be to prepare for this as well as to achieve the immediate objectives.

Young children should not be pushed to achieve correct answers in these activities if they are not ready.



TANGRAM ACTIVITIES



ANGLE ACTIVITIES

Join the angles of the same size to each other with a line.



CHAPTER 2.3: NON-STANDARD UNITS

As is stated in the OVERVIEW section, a unit is a quantity of an attribute which has to be used as a basis for

comparison in measurement. Once a unit has been chosen, comparisons can be made between this unit

and the object to arrive at a number. The number reflects the amount of the attribute possessed by the

object as compared with the unit. Unit is the way we have taken continuous measures and “discretified” them

so that we can apply number to them.

When the unit chosen varies from one measurer to another, the unit is called non-standard. Such non-

standard units are paces, arm lengths, capsful, etc. when the unit has been defined so that society in general

uses the same one, it is called standard. Such standard units are litres, metres, seconds, etc.

Units are the basis of measurement in the real world – it enables number to be used in relation to space and

time. We have to develop facility with these units – to overcome the mystique of their special names – to get

used to seeing the continuous measure in discrete-number terms. Non-standard units are an excellent way

to introduce units – they are personal and natural and allow children to see out standard units as products of

history and convenience.

The work on non-standard units is the most important section of primary mathematics as it enables teachers

to develop the processes that lie behind measurement:

the same units should be used when measuring and comparing;

the largest number means the most attribute when units are common;

the larger the unit, the small the number (and vice versa);

measurements are accurate not correct (tolerance for error);

estimation;

choosing appropriate units; and

the need for a standard.

This will ultimately lead to understandings that can control the measurement process (not just comprehend

it) – to children being able to identify and define appropriate units for attributes under investigation, to

developing instruments calibrated to measure these attributes, and to knowing the limitations of these

measurements.

This chapter, therefore, focuses firstly on the non-standard units suitable for each attribute in primary

measurement (in Unit 2.3.1) and then on the development of the processes described above (in Unit 2.3.2)

finally, special attention is given (in Unit 2.3.3) to a process which can cause particular difficulty, the process

‘tolerance for error’.

Unit 2.3.1: Non-standard units for primary measurement

Focus

This unit looks at the variety of non-standard units that can be used for length, area, volume, mass, time,

money or value, temperature and angle.

Background

Non-standard units can be anything that the measurer wishes – body parts and common objects. Some

examples for the various attributes in primary measurement are:



length – digit (finger width, or length from end of finger to first knuckle), palm (length across the palm), hand span,

cubit (length from fingers to elbow), fathom (finger tip to finger tip), pace, pencil length, blackboard duster

length;

area – any tessellating figure (triangle, quadrilateral, regular hexagon, etc.), stamps, hands;

volume – cups, glasses, any tessellating solid (cubes, pyramid, prism, etc.);

mass – paper clips, ball bearings, marbles, pens, blackboard dusters, rubber band and margarine containers;

time – clapping, hopping, counting, pulse, pendulum, candles, sand timers;

money or value – play money, shells;

temperature – column of coloured water in a

thin tube; and

angle – small paper cut out of a ‘thin’ angle,

e.g.,

Activities

Materials required: glass coffee jar, large rubber band, paper, string, margarine container, felt pen, assorted

junk materials for measuring (jars, weights, cups, solid objects, etc.), water scissors, cardboard, thin glass

pipe, red colour, containers.

1. Measure the length of the room in:

cubits

foot lengths

fathoms

paces

Questions

The length of the room has not changed so why do different people get different numbers for its length?

What does this exercise say about using cubits or paces to order a product by length over the phone?

Despite these problems, are there advantages in using cubits, paces, etc. over, say, a metre stick for measuring

the length of a room?

Can you think of real world situations where cubits, paces, etc, would be more useful than metres and

centimetres, say?

What are the commonly used standard units for length? How will their use enhance communication in length?

2. Estimate, then measure, the length of your desk in:

palms

digits or fingers

hand spans

Consider questions similar to those in 1 above.

3. Estimate, then measure, the area of your desk in:

hands

dusters

A4 pages

Consider questions similar to those in 1 above.

3. Construct a volume measurer and a mass measurer as follows.



Volume Measurer

Materials – glass coffee jar, felt pen.

Divide the height of the jar into finger widths (see

diagram). Number these lines.

Mass Measurer

Materials – large rubber band, hook, paper, felt pen,

string, margarine container.

Attach margarine container to end of rubber band with string and attach both to

the hook (see diagram). Stick the paper behind the rubber band and divide the

height of the paper into two finger widths (see diagram). Number these.

Use your volume and mass measurers to find the volume and mass of objects.

Estimate first.

4. Make a measurer for temperature and angle using the following materials:

scissors, cardboard, glass container, thin glass pipe, red colour, felt pen,

5. How would you explain to a child what a clock measures? (Note: A collection of measurers in terms of

non-standard units is given for time at the end of this unit.)

Could we have metric or base 10 units for time? What limitations would there be on any unit? Are

there any “natural” time units?

Brainstorm a number of interesting ways to measure time that children could use.

6. What can we do to develop non-standard units for Money (or value)?

7. Choose which of the following reasons for using non-standard units you feel are most valid. Justify

your choices.

Non-standard units are easier to use than standard.

Historically, non-standard units were used before standard units were established.

Children are more involved when they choose a unit than when a unit is imposed.

Children do not have to use non-standard units with the same accuracy as standard units.

It is good practice to give children experience with a variety of instruments.

By using non-standard units, children are shown that anything can be measured with some object that one

possesses.

Materials are easier to obtain for non-standard units than for standard.

Teaching hints

Ensure that children use the non-standard units “correctly”, use the same units throughout, and “align” units appropriately

(e.g., leave no large gaps when finding area, keep units in a straight line when measuring length, do not spill water when

measuring volume).

Encourage imagination in the use of units – non-standard units can be a great fallback when measuring instruments are

left behind and people become aware of a wide range of possibilities in these situations.

History can be a tremendous fallback to give interest to non-standard units. For example: As the Roman army marched,

it counted each time its left foot came down – this was called a passus.; and 125 passus (or passi) was a stadia and

1000 passus was a mile passus (which became the mile). In old England, fields were made a stadia long – their width

was determined by how much room was needed to turn the oxen to the plough. The stadia became the furrow long (and

so the furlong), the stadia became the chain, and the area became the acre.



Unit 2.3.2: Developing measurement processes

Focus

This unit focuses on how to use now standard units to develop the processes that lie behind the overall

measuring process.

Background

Measurement is comparison with units. To understand the measurement process is to understand how units

relate to the numbers that are given to the objects to measure their attributes as a result of comparison with

them. The basis of this understanding is 8 processes:

The same units must be used throughout a measurement;

Common units must be used for comparing;

Objects with the most attribute has the largest number (and vice versa);

The larger the unit, the smaller the number (and vice versa);

Measurements are accurate not correct (tolerance for error);

Skill in estimation is essential;

Appropriate units must be chosen; and

There is a need for a standard.

One of the major foci of primary measurement should be developing these processes, firstly with non-

standard units and then with metrics. But the main thrust of the development should be in non-standard units

where names, notation and conversion rates are not necessary.

Activities

Materials required: Pen, paper, Cuisenaire rods, angle pieces, pendulums, materials for investigation.

1. Cuisenaire rods can be used as follows for the measurement processes.

Process 1: Give each child a pile of different rods and ask them to place the rods end to end along

their book. Discuss the reasons for the different number of rods each child will have along their book.

Discuss which rod could be used to name the number.

Process 2: Give each child different rods and ask them to measure how many of their rods end to end

equal the length of their book. Discuss the different answers.

Process 3: Organise the children to have two objects, one longer than the other. Direct them to

measure the shorter object with rods and then add rods until this length is extended to the length of

the longer object.

Process 4: Direct the children to measure their book with two types of rod (preferably where one rod is

double the length of the other). Discuss the resulting numbers.

Process 5: Discuss situations where the length is, 4 and a bit rods. These situations emerge when

long rods are used.

Process 6: Require the children to estimate the number of rods before measuring.

Process 7: Give a variety of measurement tasks and require the children to select the most

appropriate rod for each task.

Process 8: Discuss the problems people have in deciphering the numbers when they do not know the

rod you are using.

Try these activities with your group. Can you think of other imaginative ways of achieving the same purpose?



2. Working in groups, brainstorm activities to introduce the measurement processes for the attributes and

for the materials below:

Angle – a variety of different sized cardboard sectors of a circle; and

Time – pendulums with a variety of different string lengths.

3. Investigation

At the end of this unit there are two teaching sequences of non-standard unit activities that focus on

the processes. The first of these (both were developed with the assistance of Tom Cooper and Rod

Nason by students at Riverina CAE) is on Length and focuses on the concepts of:

the larger the object, the more units needed

the bigger the unit, the fewer units needed (and vice versa)

units cannot be mixed in measurement

the same units must be used in comparison

choosing appropriate units

the need for a standard

The second of these is on Time and focuses on the processes:

the need for same and constant units for comparison

the longer the interval, the more units needed (and vice versa)

the smaller the unit, the larger the number (and vice versa)

choosing appropriate units

the need for a standard

Read these activities. Note their purpose, sequencing and use of materials. Prepare a similar

sequence of activities, for non-standard units and for the basic measurement processes, for anyone of

the following:

Area

Volume

Mass

Money or Value

Temperature

Angle

Teaching hints

The focus of the activities herein should be on the objectives described not solely on proficiency with units.

Sequences should be determined so that the process ends with the need for the introduction of metrics.

Yet proficiency with units is one of the necessary products of the non-standard unit stage and should not be

forgotten while the concepts are being tackled.

The introduction of units, the proficiency in their use and the development of the concepts, necessitates a

large amount of time (many years) of the primary syllabus being devoted to non-standard units.



Unit 2.3.3: Tolerance for error

Focus

This unit focuses on one of the measurement processes: tolerance for error. In arithmetic, answers to

exercises are exact. This is not so for measurement where the practical difficulties with instruments also

answers only to be accurate to the needs of the task.

Background

It is essential that children grasp the fast that exactness is not possible in measurement and that they learn

to tolerate error. This can be difficult to acquire and much of what we cover in this unit may not be directly

teachable to children.

When we measure something, say, the area of a tennis court, we go through four steps as follows:

We assume that the tennis court is a rectangle and that it is flat [we create a model]

We choose a tape measure [we choose a measuring instrument];

We measure the length and breadth with the tape measure (in the same units) [we apply the instrument]; and

We multiply the length by the breadth to determine the area of the tennis court [we perform the computations].

Each of these four steps has the potential for error as follows:

The tennis court may not be perfectly constructed and/or its rectangle;

The tape measure may not be perfectly constructed and/or its markings may only be to the nearest centimetre so

estimates will have to be made;

The tape could be imperfectly applied to the sides of the court, the markings could be wrongly read (there is a lot

of error in measuring something like a court); and

Round-off error will result from the computations (there are also arithmetic errors).

Measurement is a complex human activity which is doomed to inaccuracy. But in real situations, exactness is

not needed. Only measurements accurate to certain tolerances are required. For example, races are timed

only to hundredths of a second, windows are measured to the nearest mm and mass to the nearest kg.

Activities

Materials required: ruler marked in cm only, ruler marked in mm, measuring cylinders, beam balances and

masses, clock marked in minutes.

1. Measure the line below with a ruler:

Marked only in cm

Marked in mm.

What differences exist between the two measures? What is the inaccuracy of each instrument?

2. For each of the examples below, do the following:

measure each with appropriate units and instruments;

compare your answers with others in your group;

discuss in your group the four steps you used (model, instrument, application, computation); and

discuss why your answers differ (if they do) and which of the four steps was the major source of error (in your

opinion).

length of classroom wall

volume of water in a glass



mass of a bag of marbles

as directed by you lecturer

3. What is the instrument error in:

volume of a jug with measuring cylinder marked in 10 mL steps?

1-hour interval with a clock marked in minutes? and

area of a table with a ruler marked in cm (remember Step 4???).

4. Can you think of a measurement that you can exactly measure? If not, how can we reduce error?

(How can we reduce error in particular for measures of children’s ability?)

5. When would you introduce the notion of inaccuracy to children? How?

Teaching hints

The best way to introduce tolerance for error appears to be in discussing the children’s own measurement

experiences. All children will make errors when measuring. Allow them to discuss the differences that occur.

Ensure that discussion covers errors due to measuring skill and errors due to instrument limitations.

In the long run, instrument errors, particularly those related to unit markings, should be the major focus of

instruction.

A tolerance for error is essential if children are to use investigations involving measurement as a vehicle for

discovering formulae and relationships.



TIMERS

(1) Candle Clock

Materials: candles or tapers or birthday candles, pins or marker pen, tin

lids, matches, timer.

Construction: Light one candle (or taper) and use timer to help mark, with

pins or marking pen, the other candle (or taper) at suitable time intervals.

Use shorter time intervals for birthday candles and shortest time intervals

for tapers.

(2) Sand Timer

Materials: Containers, card, very dry sand (or salt), timer, scissors,

sticky tape.

Construction: Cut a circle of card and make a cone. Fill cone with sand

(or salt) and let run into container for 1 minute or another suitable time

interval (use timer). This amount of sand will then enable you to time a

minute or the other suitable time interval. (Note: can also mark the

cone to indicate level giving various time intervals.)

(3) Water Timer

Materials: Plastic container, bricks, timer, marking pen.

Construction: Pierce small hole in upper container. Fill this container with

water. Cover hole with finger. Let water run into bottom container and mark

bottom container every 30 seconds or other suitable time intervals (1 minute

or 2 minutes), or mark top container.

(4) Sinking Timer

Materials: Large bowl, containers, plasticine, timer

Construction: Pierce hole in base of container and weigh it

with ring of plasticine so it sinks evenly. Use timer to find the

level the container sinks to in 1 minute (or other suitable

time interval).

(5) Obstructed Slope Timer

Materials: Plank of soft wood (1m x 15m x 1 cm),

cardboard, marbles, timer.

Construction: Attach cardboard barriers and edges to

plank. Experiment with position of barriers and angle or

slope of the board so that it takes 10 seconds (or some

other suitable time interval) for the marble to roll down the

course.



(6) Pendulum Timer

Materials: Hook, ruler or stick (or line and weight), timer, string, weight

(washer)

Construction: Make a pendulum. Count how many swings (forward and

back) it makes in 30 seconds (or other suitable time interval). Change the

length of the pendulum to change the timing.

(7) Rolling Timer

Materials: Ruler or thin long board with groove in middle, marble, timer,

plasticine, marker pen

Construction: Set up ruler in doorway.

Use timer to find out how many full rolls (forward and back) the marble

makes in 20 seconds (or some other suitable time interval).

Place plasticine in grove at centre and roll it firmly with marble. Mark the

starting position from which the marble takes 20 seconds (and other suitable

time intervals) to come to rest on the plasticine.



LENGTH ACTIVITIES

Length Activity 1 – Measuring height

Concepts to be developed:

the larger the object the more units needed,

ordering,

betweenness

graphing

Materials: Streamers, bundling sticks, large sheet paper.

Class Organisation: Normal seating, as long as it allows for each child to be able to see front of room.

Procedure:

As an introductory lesson, this activity should begin with a class discussion on the concept of betweenness – e.g., tall, taller, tallest; short, shorter, shortest (etc). Children will be chosen to demonstrate these concepts. Beginning with one child (and gradually bringing more and more in – up to about 6), assemble a line increasing from shortest to tallest. Randomly choose children, and ask class to place child in correct position. This is developing the notions of order, betweenness, and height. This section is based entirely on questioning (e.g., Where should John stand in this line? Are you sure – why? why not?)

Using these 6 children, the teacher quickly takes a streamer measurement (height) of each child – from head to foot. Next, divide class into 6 small groups – each group containing 1 measured child. Distribute sticks and each group to measure streamer using these – length to be written onto streamer.

Class is re-united and the 6 streamers placed in order from longest to shortest by class-extensive questioning by teacher to get children to justify placements. These streamers will be pasted onto paper to form graph, the respective lengths (number of stick units) will be written on graph. Once the graph has been constructed, the remainder of the lesson consists of questioning, examples of these questions would be:

How many units long is the longest streamer?

How many units long is the shortest streamer?

Do the streamers in the middle have more, less, or the same number of bundling stick units as the (a) longest streamer, (b) shortest streamer?

What do you notice about the streamers – in particular the lengths in relation to the number of units required? Hopefully, the longer the streamer, the more units required.

What is the relationship between the length of the streamers and the height of the respective people?

Length Activity 2 – Arm lengths

Concepts to be developed: The larger the object, the more units needed, graphing, ordering. [This lesson is a

reinforcement lesson, and the concepts to be developed are the same as in the previous lesson. However, in this activity, the children are able to work on their own streamer, rather than having to share one.]

Materials: Streamers, string, pegs, large sheet paper, Unifix units, small piece of cardboard (or Post-It) per child.

Class Organisation: Normal classroom setting

Procedure:

1. Fasten a length of string on the blackboard (one side to the other for maximum room).

2. Take two streamer measurements of each child’s arm length. One is to be pegged onto the BB string , with 1 peg securing streamer to string and another peg on bottom of streamer to keep it straight; the other streamer is to be kept by children. (Names to be written on)

3. Streamers are to be ordered from longest to shortest by class (involves estimating and validating).

4. Distribute Unifix cubes and ask children to measure their own (2nd) streamer using these. When this is done, write number of units onto small piece of cardboard and pin this onto appropriate streamer hanging on the BB string. (Collect second streamer for use in a later lesson.) The piece of cardboard is added to the streamers to show the correspondence between number and length.

Key questions:

How many units long is the longest arm length?

How many units long is the shortest arm length?

Do the streamers in the middle have more, less, or the same number of Unifix units as the (a) longest, (b) shortest streamer?

Does the longest streamer have the most number of units? Do the middle streamers always have more units than the shortest streamer?

What is a rule!!!! we could make up for this?

N.B: these are only examples of questions that could be asked – the teacher must respond to all questions asked, and also must ask many questions to gain an understanding of the children’s depth of understanding. At the end of this activity, the teacher and the class should remove the streamers from the string, and paste them to paper to form a graph, which will be used later.



Length Activity 3 – Measuring our book

Concepts to be developed: The bigger the unit the fewer units are needed. The smaller the unit the more units are needed. (Inverse proportion.)

Materials:

Approximately 6 musk sticks per child (lollies) (child motivation)

2 large handfuls of Smarties per child (child motivation)

Children’s number books

Class organisation: Children seated at desks and to work individually. Class discussion.

Procedure:

1. Children to measure book from end with musk sticks (making sure that musk sticks are also placed end to end, and in a relatively straight line). Estimate before actually doing. Children will count how many musk sticks were needed.

2. Step 3 is repeated using the Smarties. Careful instruction by the teacher is required in order that children place the Smarties side by side, that is, O O O NOT O

O O. Children will count how many Smarties were needed.

3. A number of children are asked to give their results. Questions:

Were your estimations correct?

Did everybody have the same results? Why/why not?

Were more musk sticks or Smarties needed?

Why did we need so many Smarties to measure our book?

4. Children to eat lollies (motivation and reward.)

Length Activity 4 – Magnetic board

Concepts to be developed: The smaller the unit, the more units needed to measure the same object (i.e., the notion of

inverse proportion). Develop the notion that measurements cannot be compared unless there if a uniformity of units.

Materials:

Streamers. Yellow Cuisenaire rods cardboard cards – 2 per child. Orange Cuisenaire rods, graph.

Magnetic board and magnetic orange and yellow rods.

Class Organisation: Children to work individually. Seating allows for discussion.

Procedure:

1. Teacher clips a streamer to the magnetic board and demonstrates to children how to measure it with orange rods.

2. Children are given a streamer each and proceed to measure it with orange rods, counting the number of units (orange rods) long it is and writing it on one side of their card.

3. The card is then pinned under each child’s streamer on the graph. Ask: Has the longest streamer got the most orange units? Has the smallest streamer got more or less units? Why?

Repeat steps 1 – 3 using yellow rods. Ask: How many yellow rods has the longest streamer got? How many yellow units has the smallest streamer got? Why has the longest streamer got more yellow units than the other ones? Has the longest streamer got more yellow units than the other ones? Has the same streamer got the most number of yellow units? Were more orange or yellow units needed? Why? Did everyone use more yellow units than orange ones? If child says “shouldn’t we have twice as many yellow rods as orange ones”?, point out that this may not be the case as measuring is approximate and not always accurate.

Length Activity 5– Cannot mix units

Concepts to be developed: Measuring units cannot be mixed – they must be the same to enable measuring/comparing

length; estimation skills.

Materials: Part 1 1 long streamer (30 units); 1 short streamer (50 units)

Part 2 two streamers of equal length; 1 set Cuisenaire rods

Classroom organisation: Children seated in a position so as they can see the teacher demonstrating with the streamers.

Procedure:

Part 1

1. Children asked to estimate (guess) and mark a point on the blackboard where they think 30 units would be. (Brief class discussion and consensus taken).

2. Children repeat step 1 with 50 units. Would it be bigger or smaller than 30 units? Why?

3. Teacher shows the streamer 30 units long (long streamer). Does it match the point we marked?

4. Show streamer 50 units long (short streamer). Does this one match our point? Is it longer or shorter than the other streamer? Why does the shorter streamer have more units? What sort of units might we have used for each one?

5. Teacher shows children the units (i.e. a small and a large unit). Notice different sizes of units and why the shorter streamer has more units.



Part 2

Three streamers are displayed on board. Each streamer has a mixture of Cuisenaire rods measuring it (also displayed) N.B. could be displayed along base of blackboard. Children count the number of units in each streamer.

Questions

Why do we get different numbers of units if all 3 streamers are of equal length?

What did we measure each one with?

Is it fair to say that one has more units than another? Why? What or how should we have measured them to make it fair?

Length Activity 6 – Measuring our liquorice

Concepts to be developed: The same units must be used to facilitate comparison of the length of two objects (i.e., you cannot mix units when measuring the lengths of two objects). Develop estimation skills.

Materials: Long and short liquorice sticks (one of each per child) motivation! White and pink Cuisenaire rods.

Class organisation: Pupils to work individually but seating should allow for sharing of rods and class discussion.

Procedure:

1. Teacher tells children of a piece of liquorice she has that was 5 units long. Class is asked to estimate length by demonstrating with hands (N.B. Class not told the units that were used, that is, pink rods ).

2. Teacher shows liquorice and 3 different-sized rods. Children are asked to predict which unit was used to measure 5 units.

3. Liquorice (long piece) and rods are distributed. Children test prediction. Gain consensus on which unit was used by questioning children.

Key questions: Using the same unit, get class to predict how long a 10 unit, 2 unit and 40 unit piece of liquorice would be.

Tell class of another piece of liquorice that is 20 units long. Would it be longer or shorter than the first piece? Get several ideas. Show the liquorice. Why is it shorter than the first piece? How many units were in each one? What units did we use for the first piece? How many units were in each one? What units did we use for the first piece? Did we use the same units in the second piece? What size units might we have used? Why do you think this? Is it fair to say that the shorter piece has more units than the longer piece? If we used the same sized unit which one would be longer?

Children to eat liquorice.

Length Activity 7 – Measuring our foot

Concepts to be developed: Appropriate units

Materials: 1 sheet blank paper/child, scissors, square stickers. (Square rather than round stickers are used because Year 2 pupils find it easier to put together items that have straight edges rather than curved edges, and it is important that the stickers are as close as possible to each other, for example:

Class Organisation: No particular organisation required.

Procedure:

1. Ask students to trace around their own foot (with shoe on) onto a piece of paper. Cut out shape.

2. Ask children to find out how many sticker units long their foot is.

3. Ask pupils to measure their foot using their pencil.

4. Ask several children about their measurements (children will be interested to see how their measurements compare with others).

Questions:

Was one unit better to measure with than another? If so, why, or why not?

Did you have problems with “left-over” pieces? Which unit caused the most problem in this area? Why do you think this problem occurred?

If there is class consensus about the sticker being the best unit to measure feet length, ask: Why was this unit (stickers) better to use? Would it necessarily be the best unit to use? Would it be good for measuring everything? Why or why not?

Length Activity 8 – Measuring various objects

Concepts to be developed: Appropriate units for particular objects; comparing

Materials: Broom handle, pencils, rods, toothpicks, drinking straws, paper clips. (These units of measurement require less teacher-preparation time, and are easier for the children to relate to as they form part of their everyday lives.)

Class organisation: Tables should be in clusters to allow several children to be working there at the one time.

Procedure:

1. Materials are to be distributed around room, several on each cluster of tables. About five children to a table should be adequate. A list of objects to be measured is to be written on board, this list could include: length of blackboard, height of chair, width of room, length of book, length of sheet of newspaper etc.

2. Children are to choose an object to measure, and select from the available non-standard units, the most appropriate unit to measure the nominated object. Pupils are to move around the room, experimenting with as many different units as possible, and are to measure as many of the objects listed as possible.



Questions:

Ask several children what unit they found to be most appropriate for measuring specific objects (e.g., length of blackboard).

How did you come to this decision?

How long did you have to experiment before you decided?

Did you have to use every unit before you decided?

Could you look at a unit and decide whether or not it would be appropriate without having to test it?

In general, could you say that the larger the object, the larger the unit should be? Why or Why not?

Length Activity 9 – Measuring with our feet

Concepts to be developed: Use of informal units to measure, use of body parts as units.

Materials: Streamers, stencil (next page)

Class organisation: Students are to work on their own initially, and then are to discuss results in groups.

Procedure:

1. Students are to have a streamer measurement taken of their foot. (This will be the student’s individual standard informal unit – based on the size of the child’s foot.) Write name on streamer.

2. Students are to measure the nominated objects and then to choose several objects of their own to measure with their unit. The measurements are to be written onto a table with the following headings.

Object to be measured Number of units needed to measure object

3. Children are to discuss own results and then to compare their measurements with others.

Questions students need to consider may be: Is everybody’s measurement stencil the same? How many different measurements for the door, the table, the blackboard, and the teacher are there? Why should these differences occur?

To conclude this activity, a discussion and explanation of differences should occur (e.g., differences in size of foot).

Other concepts:

By doing this activity, students may gain a slightly better understanding of the appropriateness of the unit. As well, the process that the larger/longer the object the more units needed, is consolidated, and the process the smaller the unit, the more are needed, is also emphasised.

Length Activity 10 – Comparison of units

Concepts to be developed: Important to use a standard unit in comparisons estimating. Choice appropriate unit.

Materials: One large paper fish per child, 3 different-sized blocks for measuring (enough for all children), small fish for teacher’s use.

Class Organisation:

Procedure:

1. Teacher tells class of a fish that was 9 units long (9 pink Cuisenaire rods long). Class asked to estimate length by demonstrating with hands (note – class not told yet what units were used). Take several examples.

2. Teacher shows larger fish, and whilst showing 3 different measuring units (pink Cuisenaire, orange Cuisenaire and Unifix blocks), asks children to predict which unit was used to measure 9 unit fish. Fish and measuring units are distributed, children to test prediction of unit used, by testing with the three units. Gain consensus on which unit was used by questioning children.

Key questions: Using this same unit, get class to predict how long is, 3 unit, 6 unit, 20 unit, 35 unit and 18 unit fish would be – children are to justify their estimations.

Next, tell class there is also a 65 unit fish “would this fish be longer or shorter?” – get several ideas. Show smaller fish. Ask children to predict why 65 unit fish is shorter than 9 unit fish – hopefully they will state that a smaller measurement unit must have been used. Ask children whether is would be just to compare the 2 fish without giving the unit used. Teacher then to stress the importance of the use of a standard unit when comparing, by asking children pertinent questions, such as:

Why is the 65 unit fish smaller than the 9 unit fish?

Why is it important the same-sized unit is used when comparing?

If we used the same sized unit which fish would be longer?



TIME ACTIVITIES

Time Activity 1 – Introduction to non-standard units

Concepts to be developed: (1) Non-standard unit; (2) Realise the need to have the same (and constant units) for comparison

Materials: Each student will need: a copy of a workcard; a pencil; Each pair of students will need: a piece of paper

Class Organisation: Work in pairs

Procedures:

Step 1. Children work through Workcard 1

Step 1. Questioning. Refer to the comparisons made in Workcard 1

e.g., set 1. (a) Do 10 knee bends; (b) touch your toes 15 times

How can we tell which activity is the fastest (or slowest) without racing each other in the activities? [Elicit the need to time the activities.] How can we time the activities to see which is the fastest without using a clock? [Elicit: counting, tapping clapping etc.]

Step 2. Use counting, tapping and clapping to measure the three intervals of the activities on workcard.

Step 3. Fill in Workcard 1

Step 4. Questioning

How good are these (counting, tapping, clapping) for measuring time?

What if I counted like 1, 2, 3,….4….5, 6, 7, 8…9…10 etc (i.e., an irregular count) to measure one time interval and like 1, 2, 3, 4 etc (very quickly) for the second time interval? Would the counting be appropriate for a comparison? [Elicit the need to have constant/same units for comparison.]

What if we used tapping to measure the time interval for one activity and clapping for another activity? Would we be able to make a fair comparison? Why not? [Elicit: he need to have the same units for comparison.]

WORKCARD 1

Sets of activities Number of counts Number of taps Number of claps

10 knee bends

15 touch toes

Saying name & address

Writing alphabet twice

20 hops

20 blinks

Time Activity 2 – Pendulum and sand timer

Concept to be developed: The longer the interval, the more units needed.

Materials: Each child: copy of Workcard 2, a pencil;

Each group: pendulum: a ruler: a piece of wool (about 30 cm long); a small ball of plasticine; two books to rest on the ruler; a table to set the pendulum from; sand timer: two orange juice bottles; some glue; sand to fill one bottle; a piece of heavy cardboard to fit between the bottles with a small hole in it.; other tennis ball;a skipping rope;

Class Organisation: Groupsof 3

Procedures:

Step 1. Questioning

Problem: Sally wants to find out which activity would take the longest to complete – writing the word “berries” 30 times or standing up and laying down 50 times.

How could we find out which is the longest activity for Sally without having a race of the activities or using a clock? [Time the intervals by clapping, tapping etc.; make some kind of device to measure the time intervals e.g., a pendulum.] How will we be able to tell which activity is longer by using the pendulum?

Step 2 Teacher demonstrates how to set up a pendulum:

Place a ruler on the table, making sure half the ruler is balancing over the edge of the table.

Place 2 books on the ruler to keep it in place.

Roll a ball of plasticine about the size of a 10c piece and stick one end of the length of wool on the ball of plasticine.

Place the other end of the wool on the top of the ruler and place a small ball of plasticine on top of the wool to keep it in place.

The final display should then look like this:

Step 3. In groups of 3 set up a pendulum and complete the activities on Workcard 2.

Step 4. Fill in Workcard 2.



Step 5. Questioning.

Refer to the results of sally’s activities on workcard no. 2

(a) Which activity was the longest? (b) How could you tell?

Elicit: the longer activity had more swings of the pendulum.

Refer to the results of the other activities on Workcard 2.


Elicit: more swings of the pendulum were counted for the longer activity.

Step 6. Let’s test this idea – the longer the interval the more units needed - with another measuring device, the sand timer.

Teacher demonstrates how to set up a sand timer.

(a) fill one orange juice bottle with sand (b) place glue around the rim of the bottle

(c) stick the piece of cardboard on top, making sure the hole is over the middle of the bottle.

(d) place glue around the rim of the other bottle

(e) stick the bottle on the opposite side of the cardboard.

The final display should look like this:

Step 7. In groups of three set up a sand timer and complete the activities on Workcard 2.

Step 8. Questioning

Refer to the results of the sand timer activities on Workcard 2.


Elicit: the longer the activity the more units needed.

(c) Tell a story: On the plant Uropia a new unit has been worked at to measure time. This unit has been called zillfills. If we wanted to measure the time intervals of the following activities in zillfills which activity do you think would require the most zillfills?

(i) blinking 5 times;

(ii) drinking a glass of water; or

(iii) reading a 200 page story.

WORKCARD 2:

Make pendulum and sand timers as above. Complete these activities with them.

PENDULUM TIMER Number of swings Which activity is the longest? (√)

30 copies of the word “berries”

50 lying down to standing up

SAND TIMER Number of times the sand timer fills Which activity is the longest? (√)

20 bounces of ball

30 skips of rope

30 drawings of a circle in air

Reading a spelling list out aloud

Time Activity 3 – Pulse and candle

Concepts to be developed:

The smaller the unit – the larger the number of units needed.

Materials/Aids/Equipment:

Each student will need: A pencil; a copy of Wordcard 3

Each pair of students will need: 2 or 3 small birthday candles (about 6cm long); some plasticine; a saucer; matches; a small bouncing ball; a ruler measured in centimetres

Class Organisation: Work in pairs.

Activity:

Step 1. Propose a problem - How can we find out which is the smaller unit of time (a) a pulse beat or (b) a small birthday candle clock? [race, clock].

Step 2. Let’s have a race to find out which is the smallest.

Teacher demonstrates how to set up a candle timer:

place a small ball of plasticine in the middle of the saucer. Press firmly on the plasticine so that it sticks.

stick the small birthday candle in the middle of the plasticine.



The candle timer should look like this:

Teacher demonstrates how to check your pulse: (i) hold your left hand, palm upwards; (ii) place the fingers of your right hand and close together along your wrist to fell for pulse beat.

In pairs, have a race to see which is the smallest. Discuss how they knew. (e.g., when one pulse beat was counted the candle was still burning).

Step 3. Propose a second problem – Janis wishes to use the pulse beat and a small birthday candle to measure the time intervals for some activities – She is trying to find out whether the pulse beat or the birthday candle has the largest number of units for each activity.

Which unit of time, (a) the pulse beat or (b) the small birthday candle clock, will have the largest number of units each time? Note: This question is asked to determine the children’s prior knowledge of the concept to be developed.

Step 4. Let’s test out ideas about which of timers has the largest number of units by doing several activities.

Step 5. In pairs, children complete the activities on wordcard No. 3 and fill in the workcard.

Step 6. Questioning

Which unit did we find out was the smallest? [pulse beat] Which unit had the largest number of units for the first activity? [pulse beat] Was this the same for every activity? [Yes] What generalisation can we make about the number of units needed as the time unit becomes smaller? [The smaller the unit, the larger the number of units needed.]

WORKCARD 3:

ACTIVITY Number of pulse beats Number of candle steps

Counting up to 30

10 rolls of a ball

50 hops

100 breathes in and out

Time Activity 4– Which unit should I use?

Concepts to be developed: The ability to choose appropriate units.

Materials:

Each student will need: a pencil; a copy of Workcard 4

Each group will need: pendulum - a table; 2 books; a ruler; some plasticine; a piece of wool 30cm long; candle timer - a

small birthday candle (about 6cm long); some matches; some plasticine; a saucer; a piece of paper; a ruler

Class Organisation: Work in groups of 2 or 3

Activity

Step 1. Questioning

If you wanted to time how long it takes you to run around the oval which unit would you choose to time the activity: (i) a pendulum swing, or (ii) a candle timer,?(b) Why would you choose that unit?

This questioning is used to find out the children’s prior knowledge of the concept intended to be developed.

Step 2.

(a) Consider the following problem: Robert is given (1) a pendulum and (2) a candle timer. He needs to select from these the best unit to use to measure the time intervals of some of his football exercises. Note: Robert’s exercises are on Workcard 4. How can we help Robert? [Elicit: do the activities – select the most appropriate unit to time the activities.]

(b) The teacher demonstrates how to set up a pendulum – as in Time Activity 2.

(c) The teacher demonstrates how to set up a candle timer – as in Time Activity 3.

(d) Complete the activities on Wordcard No. 4 and fill in the table.

Step 3. Questioning

Which unit did you use to time Activity 1? Why did you choose that unit? [Elicit: It is the most appropriate unit.] Note: Ask questions for all the activities.

Step 4. In order to determine whether the children have grasped the idea of choosing appropriate units allow them to write down on their workcards any other appropriate units for each activity. For this, the children may need to refer back to previous work done on time to help recall the various types of units used.

WORKCARD 4

Set up a pendulum and a candle timer. [See Workcards 2 and 3.] Complete the following activities and fill in the table.

ACTIVITY Unit used (pendulum or candle) Number of units Other possible units

20 situps

60 jumps

5 fold and unfold arms

200 jogs on the spot



Time Activity 5 – Why do we need a standard unit

Concepts to be developed: The need for a standard unit.

Materials:

Each child will need: a copy of Workcard 5; a pencil

Each group will need: a piece of paper; a table, to view a clock with a second hand.

Class Organisation: Work in groups of 2 or 3.

Activity:

Step 1: Consider the following problem - Jane and Graeme wanted to measure how long it takes to do several activities and then compare their results. They decided that they would both use pendulums to measure the time intervals and ask David to do the activities for them. When Jane and Graeme compared their measurements they found out that they quite were different.

(a) What do you think caused the different results? [Elicit: The pendulums may have been different – different lengths of string and/or different weights of plasticine.]

(b) Is there anything wrong with having different results? [Elicit: A fair comparison cannot be made, reliable patterns cannot be established.]

(c) What are some other ways we can find out how long the activities take? [Elicit: Use a clock, count, clap, tap, etc.]

(d) Which of these units would allow for the best comparisons? [Elicit: The clock – seconds.]

(e) Why do you think it is the clock? [Elicit: Timing on a clock is very regular so results are reliable and comparisons are fair.]

Step 2. Show the class a clock. Allow the children to count the seconds aloud up to 2 minutes to get a feel of the regularity of timing.

Step 3. Complete the activities on Workcard 5 and complete the table.

Step 4. Compare your group’s results with other groups.

Step 5. Questioning - When you compared your group’s results with other groups’ results which unit had the closest number of units for each activity – clapping, counting, or seconds? [Elicit: The seconds, by the clock.] What do you think caused the results of the other units for clapping and counting to be so different between the various groups? [Elicit: The people in the different groups clapped differently. Some may have clapped slowly, others may have clapped quickly. This applies to the counting also.] Why then, do we need a clock to measure time? [Elicit: (i) reliable results; (ii) fair comparisons.]

WORKCARD 5:

Complete the following activities and fill in the table:

ACTIVITY Number of claps Number of counts Number of seconds

Write alphabet twice

10 stretch up-touch floors

Take off and put on shoes

Crawl under table and out



CHAPTER 2.4: STANDARD UNITS

The units that our society has adopted at present for measurement come from the metric system. This is a

base 10 system and can be related to numeration. Parts of it are also highly related and this should be used

in its teaching.

e.g., 1g is the mass of 1cm3 or 1mL of water at 4ºC

1Kg is the mass of 1L of water at 4ºC

1 tonne is the mass of 1kL of water at 4ºC.

Standard units should not just be told to children. They need to be introduced through stages so that children

can become familiar with them.

To this end, Unit 2.4.1 focuses on the four stages of introducing standard units. (1) common unit; (2)

identification; (3) internalisation; and (4) estimation. Unit 2.4.2 focuses on the metric measures and how they

relate to the base-10 decimal numeration system. To add detail to the chapter, Unit 2.4.3 discusses how

standard units can be developed for area and volume and Unit 2.4.4 how time is developed.

Unit 2.4.1: Introducing standard units

Focus

In this unit we look at the stages involved in introducing standard units in measurement.

Background

Standard units are the formal basis of measurement. They are the units used in everyday life in our society –

in industry, commerce and science. If one has a poor concept of a particular unit, or worse still, no concept at

all, it is extremely difficult to use the unit in everyday life. To be able to use the unit is to be able to estimate

with it. To be able to estimate, one must be conversant with the unit. Memorising conversion rates and

symbols and names is not enough, one must become actively involved in measuring activities.

To learn a standard unit, one must learn to think in terms of it – the unit must be used constantly without

referring back to or translating from better known units. One has to think of mass in kilograms and height in

metres and centimetres.

The development of standard units is therefore to get across the idea of standard and then to build the ability

to estimate in those units. The following four-stage process is recommended.

1. Common unit To gain experience working with the same unit as others

To appreciate the benefits of common units

To identify, construct and use a common unit

2. Identifying the standard To develop a “feel” for the relative size of the standard units

Construction of unit and/or calibration of measuring device

3. Internalising the standard To conceptualise (internalise) the standard unit with respect to

common objects (this forms a reference base for estimating)

Measurement of personal statistics and commonly used objects

4. Estimating with standard units To estimate using the standard units

To improve accuracy of estimations

Estimating and checking by direct measurement



Activities

Materials required: Cardboard, paper, pens, straws, 2m tape, metre rulers. 30m tape, centicubes, 1 litre milk

cartons, plasticine, glass bottles, glass jars, glasses, cups, jugs, matchboxes, cassette tapes, containers of

various sizes, measuring cylinders, margarine containers, string with coat hangers, rubber bands, masses,

beam balance, rice, macaroni, marbles, sand, water, coloured straws, measuring tapes (large/small), stop

watches, calculators, cm grid paper, tape, scissors, glue, pens, rulers, 1 L soft drink bottle, collection of

glasses/jars/bottles/jugs, collection of small boxes, measuring cylinders, overflow trays, plastic bags,

collection of plastic containers, string, collection of materials for weighing (sand, pasta, rice, marbles,

sawdust, etc.), bathroom scales, beam and spring balances, masses, rubber bands, plasticine

1. A piece of dowelling (about the length of an arm) is an excellent common measure to be adopted by a

class. Make up a suitable common measure for a primary class for:

Area Volume Mass Time Money or Value Angle

Temperature is a little more difficult. Is there anything we can do?

2. Complete the activities for introducing standard units for length, volume and mass.

Length

Identification

(1) Cut 1 cm pieces from different coloured drinking straws. Thread these pieces along a string in

groups of 10 of one colour followed by 10 of another colour.

(2) Using 1cm grid paper, cut ten strips that

are 10cm in length. Tape these together to

form a folding 1 m measuring strip. This

should be placed on cardboard to make it more durable.

Internalisation

(3) Use a measuring tape to measure and record your personal body measures:

Height Arm span Waist Chest

Hip Head Neck Leg

Arm Foot Length of

hand

Ankle to knee

Wrist to

elbow

Left hand Thumb Index finger

Middle finger Ring finger Little finger

(4) Find a reference length in your body which is approximately:

1 cm ……………………………………………………………………………………….

10 cm ………………………………………………………………………………………

1 m …………………..................................................................................................

(5) Mark out a 10 m distance using a large measuring tape. Determine how many of your paces

equal this 10 m. Pace the following distances, as directed by your teacher, and use this value to

convert your paces to metres:



DISTANCE PACES METRES

Length of

room

………………..

………………..

(6) Mark out 25 m. Use a stop watch to time how long it takes you to walk this distance. Use this

time to determine how long it would take you to walk a kilometre.

Estimation

(7) First estimate, and then measure the length of the objects given to you by your lecturer to

complete the table below. Estimate and measure each distance before starting the next.

OBJECT ESTIMATE MEASURE DIFFERENCE

Lecturer’s height

Blackboard’s

length

Height of top

shelf

Height of

student

…………………..

…………………..

Volume/capacity

Identification

(1) Using 1 cm grid paper, draw and cut out a net for a

cube of side 1cm. Fold and tape to make the cube.

Connect 1 m dowels to form a cubic metre. Fill the

1cm cube with water (or sand). Pour this into another

container. This represents 1mL

(2) Use cardboard to make a net for a cube of side 10cm. Tape and

fold this cardboard to make the cube. This cube is 1 L. Check this

by pouring 1 L of water or sand into it. Check that this cube holds

the same as a 1 L soft drink bottle.

(3) Calibrate a container into 100 mL levels in either of the following

ways:

Way 1 – take a glass jar and pour 100 mL amounts into it, marking the levels with tape as you

go;

Way 2 – take a 1L milk carton, cut off the top and use a ruler to divide the height into 10 equal

intervals.



Internalisation

(4) Obtain a collection of small rectangular prisms (boxes – e.g., matchbox, cigarette box, cassette

case) and pack these with MAB units to find their volume. Check by using the formulae V = L x

B x H. Try to estimate first.

(5) Obtain a collection of jars and jugs and pour 250 mL and 500 mL into them and note levels. Try

to estimate where the levels will be before pouring.

Estimation

(6) First estimate and then measure the volumes of objects as given by your lecturer. Estimate and

measure each object before moving onto the next. ESTIMATE THE VOLUME – DO NOT

ESTIMATE LENGTH, BREADTH, HEIGHT. Use a tape for volume and measuring cylinders for

capacity.

OBJECT Estimate Measure Difference

Volume

Chalk box

Shoe box

Cupboard

Under the table

…………………

…………………

Capacity

Cup

Glass

Bottle

Plastic container

……………………

……………………

(7) Use an overflow tray and a measuring cylinder, to find

the volume of objects by immersion and overflow.

Estimate first.


Lump of plasticine

Rock

Your fist

…………………….



Mass

Identification

(1) Construct mass measurers as follows:

Method 1 – a wire coat hanger, string and 2 margarine

containers (plus masses);

Method 2 – a long piece of paper, 3 rubber bands, string and a margarine container

(calibrate the ”spring balance’” with masses – mark lengths on the paper).

Use these measurers to make up plastic bags, or other containers, containing 100 g,

250 g, 500 g and 1 kg or various materials (pasta, sand, marbles, rice, etc).

Internalisation

(2) Use a bathroom scale to measure your own mass in kg.

(3) Measure 1 L of water

(4) Find objects in the environment that measure approximately 1 kg, 500 g, 250 g, 100 g, 50 g and

1 g. Make up lumps of plasticine to these measures.

Estimation

(5) Estimate first and then measure the masses of the following objects as given by your lecturer.

Complete estimates and measures of object before moving onto the next.


Duster

Case or port

Shoe

Another student

Lecturer

………………..

………………..

3. Make up a series of activities to take children through identification, internalisation and estimation for

standard units for one of the following:

Area Time Money

Teaching hints

These activities need a laboratory set up – materials, tables, water. This requires the same attention to

safety that is required in science experiments.

It also requires the development in children of the expectation that learning will occur from what they do and

from other children as well as the teacher. Investigation style activities do not work where children have to

show everything to the teacher and be shown how to do everything by the teacher.



Unit 2.4.2: METRIC MEASURES AND DECIMAL NUMERATION

Focus

This unit focuses on metrics, the standard units for length, area, volume, mass, temperature and angle. It

shows these can be interrelated with decimal numeration.

Background

Schools use a simplified version of the metric system, the SI system of measurement. SI is an abbreviation

of Systeme Internationale (International System). It is a base 10 system of 6 base units (metre, kilogram,

second, ampere [electric current], Kelvin [temperature] and candela [luminous intensity] agreed to at an

international conference in 1960.

Primary schools use only the SI base units of metre, kilogram and second. All other SI units are derived,

e.g.,

a centimetre is 1/100 of a metre,

a cubic centimetre is a cube with sides of 1 centimetre,

a litre is 1000 cubic centimetres.

Once a name has been fixed for a SI unit (e.g., metre for length), larger and smaller units of that

measurement are made by putting a code word, or prefix, before that name (e.g., centimetre, kilometre) –

like a given name and a family name. The prefixes mean the same for every “family”. SI has two prefixes for

all units (kilo – meaning “a thousand times” and milli – meaning “a thousandth of’”) and, and for length only,

centi – meaning “a hundredth of”.

SI units in the primary school

The SI units that are used in the primary school are listed in the Overview section. However, when writing

the symbols, there is a space but no full stop and no plurals (no “s”) are used; capital letters are used for litre

only (in primary schools) (e.g., correct – 40 kg; incorrect – 40 Kg., 40 kgs or 40kg). It should also be noted

that a comma is no longer used as a thousands separator, a space is used instead (e.g., correct – 49 216;

incorrect – 49,216), and a zero should always be placed before the decimal point to denote that there is no

whole unit (e.g., correct – 0.53; incorrect – .53).

Activities

Materials required:

Metre rules, measuring tapes, measuring cylinders, cm grid paper, metric masses, beam balances,

thermometers, protractors, stop watches, timers, calipers, cm cubes.

1. Obtain examples of measuring instruments for metrics and measure the objects below:

length of room – metre stick,

length of blackboard – measuring tape,

volume of coffee jar – measuring cylinder,

area of blackboard duster – cm grid paper,

mass of blackboard duster – metric masses / beam balance,

mass of blackboard protractor – masses / beam balance,

temperature of water – thermometer,

angle between index and middle finger – protractor,

time to write signature 10 times – stop watch,



how many hops in 15 seconds – tocker timer,

width of can – callipers, and

volume of small packet – cm cubes.

Estimate first in all cases.

2. Metric Expanders

(a) Copy the four number expanders (or construct larger copies) at the end of this Chapter and cut them

out. Use them to construct metric expanders as follows:

Expander A – metres, centimetres and millimetres,

Expander B – kilometres, metres and millimetres,

Expander C – tonnes, kilograms and grams, and

Expander D – litres and millilitres.

(b) Fold the metric expanders like number expanders. Use them, as directed by your lecturer, to relate

metres, kilograms and litres to the other units.

Note: The decimal point has been placed so that Expanders A and B read as metres, Expander C as

kilograms and Expander D as litres when folded.

(b) Construct an expander for hectares, square metres and square centimetres (which reads as square

metres when folded).

3. Metric Slide Rule

Copy the metric slide rule at the end of this Chapter (metric slide rule is an idea from Baturo A and English L,

Sunshine Mathematics, Melbourne, Longman/Chershire, 1985/86). Using scissors, cut out the slides and the

scale and slit the scale along the dotted lines. Then, using the rounded end of the slide as a tongue, thread

each slide from the back up through the slit on the left of the scale and across the front and out the slit on the

right of the scale.

Use the slide rule to relate metrics and decimal numeration.

4. Time is often represented as 2.43 hours. What is the numeration difficulty here? What can we do to

diminish this difficulty?

Teaching hints

1. Give children experience with specialist measuring instruments such as callipers, verniers, etc.

2. Undertake outdoor activities such as height measurement, orienteering and scale drawings/surveying.

3. Metric conversion rates need to be consolidated through drill – some examples

(a) Dominoes

(b) Bingo

c) Mix and Match cards



(d) Card decks (for concentration, gin rummy, snap, etc.)

4. Metrics should be introduced along with decimals. They apply decimal understanding and reinforce

decimal concepts. For instance:

two decimal places are related to money (dollars and cents) and length (m and cm); and

3 decimal places are related to length (m and mm), mass (Kg and g and t and Kg) and volume (l and ml).

The only problem is time which is base 60.

Unit 2.4.3: Area and volume

Focus

In the development of standard units, there are techniques and materials that enable children to experience

measurement with out the need to apply formulae. This unit looks at particular techniques and approaches

useful in Area and Volume. It also offers activities that can be part of the development of the process of

measuring Area and the development of the process of measuring Volume.

Background

The sequence of activities to standard units involves

identifying the attribute,

comparing and ordering, and

non-standard units.

Hence, when you begin to teach children about area, for example, you should give preliminary activities,

which focus on covering and comparing the different amounts of cover needed for different objects. It is also

useful to do some activities with non-standard units. It is only at the end of such activities that the

experiences included in this unit are meant to be presented.

It is also important to remember that measures are not exact. This is particularly important for area and

volume work where squares and cubes are counted. The numbers that result are only approximations for the

areas or volumes.

Volume is a particular difficulty in that there are two units of measure – cubic cm for ordinary volume and mL

for capacity (liquid volume). This dichotomy is exacerbated by the differences that exist in the difficulty of

each unit. Capacity is simple and quick to measure with the special measuring cylinders and cups. Children

can pour from one container to another to do this at an early age although conservation does not exist to

late. The computations involved in the rest of volume make it a topic for the upper years.

Capacity through the use of measuring cylinders is much easier for irregular shapes that can contain sand or

a fluid. Computations that give rise to ordinary cubic measures are easier for regular shapes for which a

formula is known.

The two models of volume are related – a cubic cm is a mL and 1 000 L is a cubic m.

Activities

Materials required



Square grids (at end of unit), tracing paper, pens, cm cubes, measuring cylinders, collection of classes and

jars, cm cubes, Cuisenaire rods, water or sand, geoboards and rubber bands (dot paper), three grids (2cm,

1cm and 2mm).

1. Beginnings

What is area? How would you describe it to a child? How would you decide which of two objects has more

area?

Which of the three shapes below would be best for finding the area of this page? Which would make the best

unit?

Which of these three shapes covers best? How well do copies fit together? Which is common and easy to

draw? Which is easiest to count when copies of it have covered a shape? Experiment with a few copies of

each shape!

2. The adoption of the square

For historical and practical reasons, the square is our society’s unit for area. Hence, to find area, we need to

determine how many squares of a certain size cover a shape.

Use three grids of different sized squares (2 cm, 1 cm and 2 mm)Copy the following shapes

onto tracing paper and use the grids to find their areas. [Note: this can also be done by copying

the grids onto plastic and placing the grids over the shapes.] ESTIMATE first! Put results on a

table with headings – Shape, Estimate, Number of 2 cm squares, Number of 1 cm squares,

and Number of 2 mm squares.

Do the same for the following shapes, but in each case determine the number of squares that are completely

inside[I], the number of squares that are completely and partially inside [P] and then the average between

these two numbers [A]. ESTIMATE first! Put results on table with headings – Shape, Estimate, Number of 2

cm squares, Number of 1 cm squares, and Number of 2 mm squares.,



For each of the three grids for G above, what is the error in taking the average? In terms of 1 cm squares, which

grids, gives the least error? [Remember that each 2 cm square is four 1 cm squares and each 2 mm square

is one twenty-fifth of a 1 cm square.] Hence, which square is the most accurate? How does this compare with

which square is the easiest to count?

3. Removing tedious counting

To become increasingly accurate in area, we have to use finer and finer grids. But then the counting

becomes tedious. In come cases, the accuracy required demands too fine a grid. Hence, the invention

of formulae to give relief to the calculation of area for regular shapes. Irregular shapes still require the

counting of squares, but the invention of calculus enables this also to be achieved without tedious

counting.

How fine a grid would you need to be within 1% of the correct area of an irregular shape by counting

squares and averaging (as we did in 2 above)?

4. Using geoboards

Geoboards are a useful aid for area measurement. The nails enclose squares and rubber bands can

be used to outline the shapes. There are two particularly useful techniques for polygons as are

described below.



Use the most appropriate of the above two methods to determine the area of the following shapes.



Which of the following shapes is it possible to make on a geoboard?

A – Rectangle: area of 1, perimeter of 4 B – Rectangle: area of 3, perimeter of 8

C – Rectangle: area of 4, perimeter of 8 D – Rectangle: area of 4, perimeter of 10

E – Triangle: area of 3 F – Parallelogram: area of 4

G – Trapezium: area of 5 H – Pentagon: area of 8

I – Hexagon: area of 8 J – Octagon: area of 8

5. Variation on the geoboard

Other shapes can be used for the unit of area, e.g., triangle units – the small right triangle made by

placing a rubber band around three adjacent nails:

What is the area of the figures below in triangle units.

Is it possible to construct a pentagon with an area of 11 triangle units.

6. Counting cubes

Resisting the use of any formulae you may know, count the number of cubes that could be packed into the

following 3D shapes. Try to be systematic in your counting!

Use cm cubes (or Cuisenaire rods) to estimate the volume of a can or jar.

A box is 2 m by 3 m by 40 cm. What pitfalls might children fall into in calculating its volume? What would you

suggest be done with a child who comes up with the answer of 240 cubic units?

7. Capacity

Obtain a clear cylindrical jar or glass as your unit of capacity. Use a ruler to divide it into tenths of units. Use this

unit to measure the capacity of three containers. Estimate first!

Discuss in your groups strategies to determine the volume of an irregular containers of any size. Also discuss

strategies for finding the volume of an irregular solid object of any size.



Teaching hints

The geoboard is a flat (usually square board) from which nails or pegs protrude. The nails are in a regular

array (rows and columns). Coloured rubber bands are used to outline shapes. Geoboards are useful

teaching aids for multiplication, geometry and measurement. They are particularly good for children to use in

investigations because shapes can be quickly made, tested and remade. They do have problems if discipline

is a difficulty as their rubber bands can be dangerous if used other than on the boards. The book (part of this

series) Geometry: Space and Shape in the Primary School has many activities for the geoboard. A geoboard

can be replaced with dot paper, ruler and coloured pens – it isn’t so flexible but a permanent record is

available and there is less danger of discipline problems.

There is a formula (Pick’s Theorem) for relating the area inside a polygon on a geoboard with the number of

nails inside [I] and the number of nails on the boundary [B]. AREA = I + B/2 – 1. This formula is an

excellent topic for an investigation. Direct the children to attempt to find a formula when there are no nails

inside the shapes, then when inside and so on …. . Then combine the different formulae for all these cases

into one formula for all cases.

Children require experience counting squares and cubes and measuring capacities if they are to understand

what lies behind formulae. Even though it is slow and messy, experiments of this type should be an

important part of measurement.

Measurement is a great opportunity to get outside and into everyday activities. Try to find local situations

where measures are taken and take your children to them.

Unit 2.4.4: Time and time-telling

Focus

This unit focuses on a sequence for teaching time. Because of its central place within everyday life, time

requires instruction on the concept of time and the reading of common instruments for measuring time, e.g.,

clock faces. In fact, this unit will separate the topic of time into two areas: the concept of time and time-telling

skill.

Background

The development of standard units for time (hours, minutes, days) requires the acquisition of knowledge of

time-telling, i.e. how to “tell the time”. A suggested sequence for teaching time is as follows:

1. Sequencing events;

2. Cycles of events;

3. Associating events with times of day;

4. Telling time on the hour;

5. Movement of hands on clock

5 minutes intervals,

minutes after the hour,

digital notation, and

other conventions (e.g., ‘half-past’, ‘quarter-to’, etc.);

6. Passage of time (calculating time intervals); and

24-hour time.



Because of the need to develop the ability to read a clock face, standard units and time-telling has to be

introduced early to children. In fact, some educators recommend that time-telling (the ability to read a clock

and to set a clock) be taught separately to the concept of time (the understanding of various units of time,

their duration and their relationships to events). The identification of the attribute of time and comparing

times has to occur before the introduction of these standard units and time-telling, but the main non-standard

unit work on the measurement processes can be undertaken after this standard unit of time-telling

instruction.

People make continual references to time in everyday life. The day’s program is determined by time and the

measured with the use of clocks and watches. Some of the language associated with time is confusing. As

stated by an article central to this unit [Nelson, Glen, “Teaching time-telling”, The Arithmetic Teacher, May 19

1982], the following all describe the same time:

Forty-five after six

Forty-five past six

Fifteen till seven

Fifteen to seven

Quarter till seven

Quarter to seven

Quarter before seven

Six forty-five

The understanding of time, therefore requires the simplification of this language and, because of its analog

form, the use of the tactile and kinaesthetic senses as well as the visual and auditory senses.

Activities

Materials required: Materials for construction of clock faces, geared clocks, pen, paper.

1. The non-geared clock face.

Construct a clock face – a circle, twelve markings, the numbers 1 through to 12, two hands.

What knowledge can the construction of such a clock face assist children to acquire? What questioning will assist

this acquisition?

Move the hands of the constructed clock so that the little hand is directly at the 4 and the big hand is directly at

the 6. is this half-past 3 or half-past 4? Can a real clock face ever show these numbers? What then is the

problem of non-geared clock faces as a teaching aid?

What teaching activities is non-geared clock face useful for? [Remember the materials language

symbols model for determining appropriate instructional activities.]

Use your non-geared clock face to set a clock for:

a quarter to 4

4 o’clock

6:55

25 minutes past 8

17 minutes to 11

2. The geared clock face

How can you use the gearing to help introduce the fact that at the o’clock, the big or “minute” hand points to the

12?

How can you use the gearing to introduce “clockwise” and “anti-clockwise”?



How can you use the gearing to introduce the role of the “minute” or big hand?

How can you emphasise the use of tactile and kinaesthetic senses in this teaching? [Think of ways that the

children can focus on feeling where the hands are and on “acting out” the action of a clock face!]

What activities are geared clock faces most useful for?

3. Digital clocks / 24 h clocks.

Most clocks are digital. The numbers on a digital clock can be read as soon as numbers can be red,

but there may be little understanding of what the numbers mean in terms of time. For the purposes of

teaching, digital representations of time can be considered as the symbols in the materials

language symbols model of instruction. Hence digital representations of time can be introduced in

relation to clock faces (the materials) and then later related to language (statements like “half-past 4”,

“25 minutes to 6”, “7 o’clock”, etc.).

(1) The triangular model on right shows that six

activity types should be included in a teaching

session. Complete the table below by continuing

the labelling of activity for each type. Write a

teaching-time activity for each of the 6 activity

types.

(2) 24 h clocks are becoming common in digital form. How do we amend the activities in (1) above

to make them applicable to the 24 h clock? What particular difficulties does the 24 h clock have

for children? [Remember that children are used to working in base 10!]

4. In your groups, prepare an activity for steps (1), (2), (3) and (6) in the suggested sequence for

teaching time given in the background to this unit.

Teaching hints

Let the children’s experiences, especially with geared clocks, do a lot of the instructional work for you.

Geared clocks always have the ‘minute’ hand pointing at the 12, for instance, whenever the ‘hour’ hand

points directly at a number. Children can be lead to discover this.

Collect old non-working clocks for your classroom. The gearing should still work.

Have the children act out the actions of a clock. For instance, their hands can become the hands of a clock.

Have the children shut their eyes and feel where the clock hands are at various times. Remove the glass

from the front of the clocks to allow this to be done. The kinaesthetic and tactile senses are essential in the

teaching of time.

Do not forget the concept of time. Children need experiences with time intervals, comparing lengths of time,

timers of all sorts that use non-standard units and particularly the relating of times with parts of the day –

start of school, end of school, dinner time, bed-time, etc.



METRIC EXPANDERS



METRIC SLIDE RULE



CHAPTER 2.5: FORMULAE

Once standard units have been acquired, measures can be calculated by directly comparing the unit with the

object – in most cases by seeing how many copies of the unit fit into the object. This can be a slow and

tedious process, therefore, to make it simpler, ways have been found to speed up the process. One such

way in formulae.

For most attributes, technology has been the means by which measurement has been made easier. For

example:

Mass can be quickly measured by weighing machines;

Length can be easily (though not so quickly) measured by tapes and surveying devices;

Time is quickly and very accurately measured by clocks;

Temperature is measured by thermometers and other more sophisticated devices; and

Angle is measured by protractors and surveying equipment.

Value (or how much money something is worth) is not so technologically based and area can be very tedious

to measure. Volume of liquids (capacity) can be easily measured by instruments (e.g., petrol pumps)

although volume in other situations is difficult.

But volume and area are related to length. And for regular shapes this relation can be enshrined in formulae.

Hence area and volume can be calculated from more easily measured length. Similarly, formulae for

perimeter has also been developed. The first unit in this chapter (Unit 5.1) focuses on these formulae,

looking at how they can be developed in children. It looks particularly at two teaching approaches or

strategies that can be used with children to enhance understanding and memorisation of the formulae:

discovery and relating to known formulae.

Measurement, being the application of number to the world, is the main area of applications in mathematics.

Most routine classroom problems arise in money or measurement as do many of the real problems that

children can be directed to. The second unit in this Chapter (Unit 5.2), therefore, focuses briefly on

applications and problem solving (problem solving is the total focus of another book in this series).

Unit 2.5.1: Developing formulae with understanding

Focus:

The focus on this unit is on the development of formulae understandings in children.

Background:

The formulae which is the focus of primary mathematics is within length, (perimeter) area and volume. The

most common formulae that have to be acquired are as below.



Activities:

Materials required: Geoboards, rubber bands, pen, paper, scissors, tape, glue.

1. The first strategy for children acquiring formulae is discovery. For example:

(a) Construct 6 different rectangles on your geoboards.

(b) For your rectangles, draw up and complete a table with headings – Rectangle, Length, Width,

Number of squares (area).

(c) Looking at your table, can you see a pattern for relating Area to Length and Width? How would

you use your relationship to find the area of a rectangle of length 30 and width 40? What

questions would you ask to ensure children saw the pattern above?

2. The formulae for circumference of a circle can be found the same way. For example:

(a) Measure radius and circumference of several circles.

(b) Record this information on a chart with headings Radius (R), Circumference (C), and C/R (using

a calculator..

(c) Can you see a pattern?

What difficulties would children have with measurement that may make their C/R computations too

inaccurate. How could we overcome this?



(For example – one group measured cylinders. Ten cylinders were placed side by side as

below, measured and this divided by 20 for the radius.

String was then wound around them ten times, measured and this divided by 10 for the

circumference).

3. Which is the best way to discover the formula for volume of a rectangular prism?

Way A – fill various prisms with blocks and record this on chart with headings L, W, H and Volume -

look for a pattern.

Way B – construct with blocks various prisms of (i) height one (look for a pattern – relate resulting

volume to area of base), (ii) height two (again relate to area of base), and (iii) height 3, then height 4,

and so on (attempting to relate volumes to the product of the height with the area of base).

Note: The area of a rectangle can also be discovered using way B above. It can be seen as a natural

consequence of counting one row, then a second row, and third, … up to the height.

4. Describe a way you could discover the formula for the perimeter of a rectangle.

5. Sometimes discovery is assisted by simple activities. For example, a cardboard circle could be rolled

along a line for one complete revolution, as below

It could then be shown that the length of this line is just a little more than 3 diameters or 6 radii.

Would this help Activity 2 above? Would it suffice for primary knowledge of circumference of a circle?

6. The second strategy for children acquiring formulae is relating to known formulae (schematising).

For example, work through the MAKING CONNECTIONS activity at the end of this Chapter.

How does this activity help children to remember formulae? What insight does it give into shapes

and their properties?

7. The formula for area of a circle can be got from the formula for circumference of a circle. The

formulae for volumes of cones and pyramids can be get from the formulae for the volumes of

cylinders and prisms. Do some research and find out how these are done.

8. The results from area formula must be related to the number of squares that cover the shape.

Hence, formula must be checked.

(a) Find the area of the following triangles by counting squares, and (ii) the formula for the area of a

triangle.



(b) Repeat the above for the following parallelograms.

9. Why would children think the two shapes below have the same area? Which is more useful for

showing they do not – formulae or counting squares?

10. In these activities, we have used paper and geoboards to embody area. What are the relative

advantages and disadvantages of each material? In 2 above, an activity to discover the formula for

the circumference of a circle was described. A similar method could be used for the formula for the

area of a circle. Squared paper could be used and the results (for different circles) recorded under

the headings – Circle, Radius (R), R2, Area (A), and A/R

2. What would be the difficulties with this?



Teaching hints:

Discovery and schematising activities are the basis of good teaching because they enable children to

structure their own knowledge in a form it is easy to recall from and reconstruct particular knowledge from.

Read the overview of this book (the approaches to teaching section).

Formulae can be used to practice decimal operations. For example -Addition of decimals – perimeter of a

rectangle, and Multiplication of decimals – area of a rectangle.

Seeing pattern and generalisation is an activity that should be left to the upper primary years. Even children

at this age may have to be “led by the hand” through discovery. Do not expect abstract generalisation.

There is a tendency for children to launch into formulae and to ignore the meaning of numbers generated by

the formulae. For example, it is important that children first visualise area in terms of coverage with unit

squares. Only when this covering concept is established, should the counting process be organised into the

L x W formula for rectangles. Once this has been achieved, the formulae for the parallelogram and the

triangle can be established. Then the coverage visualisation must be used to ensure that children realise that

the numbers that come from the formulae are related to the number of cm squares that cover the shape.

Hence, the importance of the checking procedure, where numbers from formulae are related to the actual

coverage in squares.

Unit 2.5.2: Applications and problem solving

Focus

This unit looks at measurement in relation to the other topics in primary mathematics and focuses on its role

in providing applications and problems for knowledge of number and shape.

Background

Problems are divided into a continuum routine (bases on mathematics content) to creative (requiring

mathematical thinking and little content base). Measurement problems are routine in the they use

mathematics content but can be sufficiently complex in their setting that they require creative though as well.

As such, they seem to lie mid-way in the routine-creative continuum. Therefore, measurement problems are

applications of number and shape that require problem-solving skills for both routine and creative problems.

Hence, they require:

good mathematics content knowledge (number, shape and measurement);

a positive attitude, self concept and attribution;

a good plan for attacking problems;

a wide repertoire of problem-solving strategies;

good thinking skills; and

good executive processes (for monitoring, planning and evaluating problem-solving progress)

Of particular importance to measurement are the following problem-solving skills.

(1) Polya’s four stage plan of attack

SEE – understanding the problem,

PLAN – develop a plan to go about solving the problem,

DO – solve the problem, and

CHECK – check the solution and look back to see what can be learnt from how the problem was solved.



It is crucial that a measurement application be understood before any attempt is made to solve it.

Likewise, the complexity of measurement application requires that they often be solved in steps. This

necessitates that an overall plan is developed before the solution is attempted.

(2) Problem solving strategies that help with understanding and enable problems with more than one

step to be tackled. Such strategies include (see Meiring, S. Problem solving … A basic mathematics

goal, Palo Alto, Cal., Dale Seymour, 1980):

Identify given, needed and wanted information,

Restate the problem in your own words,

Make a drawing, diagram or graph,

Identify a subgoal (break the problem into parts), and

Work backwards.

The use of pen and paper to make drawings and to list the different parts that have to be completed

is crucial to these problems.

(3) Visual, flexible and creative thinking and the ability to both monitor what is happening at present and

to keep in mind what has to happen next (and where what is happening now fits into the total

solution).

As always, it has to be kept in mind that problem solving is a two edged teaching tool. It can be used

to apply knowledge. It can also be used to teach that knowledge in the first place. The came problem

in one instance could be given for the processes that have to be experienced to complete it – as a

vehicle to apply old knowledge and improve thinking. In another instance, that same problem may be

given for its answer. It may, for example, be the approach used to get the children to discover a

formula.

Activities

Materials required:

Pen, Paper, any material that you think may help with the problems, your imagination.

1. 1. Measurement problems should be related to the everyday world of the child. For instance, children at a school

beside a road and next to a river could be set the following measurement problems:

How many trees worth of paper does our school use in a year?

How many tonnes of metal pass our school on the road in a week?

What volume of water flows down the river in a fortnight?

2. How would you go about determining the amount of ink (ball point pen ink, felt pen ink, typewriter ribbon ink, etc.)

that is used in your Institution in a week? Don’t work the problem out. Just list the things you would have to do to

solve this problem.

3. Are there any investigations that you would have to do in order to solve the above problem? [Remember you may

have to work out what the average ink use is in a student’s pen!] Do these investigations add to the problem?

What extra difficulties do they add to a teacher in a classroom? Are they worth it?

4. Consider that you are to take a class of children in measurement problem solving at you Institution. In your

groups, develop 5 measurement problems that could be set for these children that use the environment of your

institution.

5. Measurement problems can be extensions of measurement instruction. For example, try these problems.



What is the largest possible square that can be constructed completely inside the octagon below? (The square

may touch the boundary of the octagon.)

What is the area of the square region below?

What is the volume of the stairwell on the right? The two doors are 3 m high and 1 m wide. Each step is 0.3 m

high and 0.3 m deep.

What can the floor space of the following addition to a home be? The ceiling has to match the rest of the house

and be 3.1 m high. The airconditioner is only adequate to cool and extra 200 cubic metres of space.

6. Measurement problems can be used to introduce, or to give a reasonable appearance to, formulae. For example,

the volume of a prism/cylinder is a product of the base area and the height. Can you devise a measurement

problem for investigation that might make this formula reasonable to a child for the shapes below. [For example,

stacking coins may help for the volume of the cylinder!]



7. The MMP Measurement book alluded to in the acknowledgements of this book contained a collection of

measurement experiments. They provide examples of good measurement application and problems. Try some of

the examples below.

What is the relationship between the height a ball is dropped from and the height it bounces to? Does this

relationship vary with the ball? [Experiment with different balls.]

How could you find the mass of a suit made from 3 m by 1.5 m of material from the scraps left over?

What is the effect on the volume of a cube when you double the length of it sides? What is the effect on the

volume of a sphere when you double its diameter? [Experiment with cubes made of blocks and plasticine

spheres. Find the volume by counting blocks or by immersion in water (or by mass?)]

What measures best answer the question “how big are you?”? Do any of these measures correlate? [Plot

results for a number of people on graph paper.]

What is the relation between the volume of a cylinder and the volume of a cone? [make a cylinder and cone of

the same height and diameter out of cardboard and check.]

What is the volume of water wasted in 24 h by a dripping tap?

What is the surface area of a tennis ball? [Do not use the formula.]

What rectangle of perimeter 36 cm has the largest area? What 2D shape of perimeter 36 cm has the largest

area? [Use a geoboard. Tie a loop of string or perimeter 36 cm and make the different shapes with this.]

Teaching hints

Measurement problems are an excellent opportunity for out of doors work, for investigations in the library, for

experimentation and for open ended activity (where each child does as much as they are able).

Measurement problems allow different abilities to be catered for. Use them particularly when catering for

talented children. Their open ended nature allows different children to perform at different levels.

Measurement problems do require flexibility of the teacher. Fixed times and timetables are difficult to work

within. There is also a need to organise for many messy materials to be used. Safety is also a problem.

Cleaning up, a time consuming activity, must also be planned for.

Use the ideas of the children. Children can often think up excellent investigations. Do not be afraid of leading

children to a situation, say a boast on the river, and requiring them to make up their own problems for

answering.

Do not help children having difficulties by answering the question. Rather direct them to an experiment they

can perform that may throw light on the question.



MAKING CONNECTIONS

You need: Paper, scissors, tape, pen.

Activity:

1. Cut a rectangle out of paper. By cutting and taping, from two triangles as below:

Are these two triangles the same? How does their area relate to that of the beginning rectangle? Is

the base length and width of the triangle equal to the base length and width of the rectangle?

2. Cut a triangle out of paper. Cut a copy of it. By cutting and taping, form a rectangle as below:

How does the area of the rectangle relate to the area of the triangle? What about vase length and

width?

3. Cut a triangle out of paper. Cut a copy of it. By cutting and taping, form a rectangle as below:

What is the relation between triangle and rectangle here in terms of area, base length and width?

Questions:

(a) The area formula for a rectangle, as below, is PXQ

(b) What is the area formulae for the related triangle?

(c) What are P and Q in this triangle (do a drawing)?

(d) What about the formula for a parallelogram?



SECTION 3

SEQUENCING THE TEACHING OF MEASUREMENT

This section has two chapters. Chapter 3.1 takes each attribute (length, area, volume,

mass, time, temperature, angle and value) in turn and summarises activities that could be

given in each of the 5 stages from Section 2. Chapter 3.2 discusses the teaching of

measurement looking at approaches to teaching, limitations to learning and diagnosis

and remediation.



CHAPTER 3.1: SEQUENCING MEASUREMENT ACTIVITIES

This Chapter describes activities that can be used in primary classrooms to teach the measurement topics of

length, perimeter, area, volume, mass, time, angle, money/value and temperature. The activities in each

topic have been organised under the 5 stages of measurement. As such, they supplement the earlier writing

in this book.

The chapter is divided into four units. Unit 3.1.1 looks at length and area activities, Unit 3.1.2 looks at volume

and mass activities, and Unit 3.1.3 looks at the remaining attributes: time, money, temperature and angle. A

separate unit, Unit 3.1.4, has been provided at the end for activities that relate attributes and that can be

basis of workstations.

It is the contention of this book that effective teaching of measurement requires knowledge of the stages

through which measurement develops and a rich variety of motivating and practical activities. This Chapter

exists to increase the richness of activity available through this book and to provide a view of the stages

within separate attributes.

Delineating activities for the five stages

Stage 1 – Identify the attribute

It is difficult to differentiate ‘identify the attribute’ activities from ‘comparing and ordering’ activities. In

practice, this need not be done and both types of activities can be taught together. It should also be noted

that activities to identify particular measurement attributes should follow rich experiences with general sorting

and classifying activities and much discussion of more general attributes, such as colour, sound, etc.

There are two general ways to introduce any attribute. They should work for every measurement topic. They

are:

providing examples where the only thing that is the same is the attribute, and

providing examples where the only thing that varies is the attribute.

Stage 2 – Comparing and ordering

Comparing means two instances. Ordering involves 3 or more instances. Te process of ordering is based on

comparisons. In this handout, ordering activities are interrelated with comparison activities.

The classroom activities under this section can be divided into three types: direct comparison of two

instances of an attribute; indirect comparison of two instances of an attribute; and ordering, directly or

indirectly, more than two instances of an attribute. The activities involve no units and no numbers. The total

amount of the attribute present is compared or ordered in one action.

Stage 3 – Non-standard units

Non-standard units have two central uses: introducing the notion of the unit; and introducing the eight

measurement processes (see page 48 of this book). They also are useful in introducing children to the

appropriate techniques for using units and measuring instruments. The information if this handout will focus

on materials that can be used for non-standard units. It will not focus on the teaching needed to draw out the

uses to the same extent.

Stage 4 – Standard units

Similar to the non-standard units, this section will focus on instruments and techniques. It should be

remembered that these units should only be introduced after the need for a standard has been determined

and it is recommended that they be preceded by the use of a class chosen common unit (if appropriate). It is

also recommended (page 78 of the book) that the first experiences with the unit involve identifying the unit



through experiencing it or constructing it. Then the unit can be used, firstly on everyday things to facilitate

internalisation and, secondly on everything to develop estimation skills.

Stage 5 – Formulae and applications

The focus here will be on activities and ideas to enhance discovery and problem solving. Situations involving

two or more attributes are the most fruitful.

SPECIAL NOTE: Both time and money/value differ from other attributes in that basic standard unit skills

have to be developed early. For time, time-telling skill for analogue timers (i.e., the clock face), and digital

timers, should be taught early and may well precede work on the concept of time at the comparing and

ordering and non-standard units stages. For money/value, money-handling skill (recognition of coins and

notes and computation with money) may well precede work on the concept of value, on bartering (the

comparing and ordering stage) and on non-standard units.

Therefore, for the time and money/value sections of this Chapter, time-telling skill and money-handling skill

will be the first section covered.

Unit 3.1.1: Length and area activities

Focus

This unit focuses on classroom activities for length, perimeter and area.

Background

Length and perimeter are the measure of one dimension and area is the measure of two dimensions. As

such, length and perimeter are associated with strips of various types (e.g., rulers) and area is associated

with covering and planes (e.g., graph paper). Of course, there are other ways than the obvious to measure

each, e.g., trundle wheels to measure length, and it is hoped that a rich and varied mixture is given below.

Activities

Materials required: Pen, paper, a library or collection of resources, the materials that are mentioned in the

activities.

LENGTH AND PERIMETER ACTIVITIES

Stage 1 – Identify the attribute

Body actions, mimes and dances that represent trees and plants growing higher or fish growing longer etc.

Discussing heights, cutting strips of ribbon to children’s heights and sticking on walls, marking heights on

walls, drawing around a child lying down, drawing around a child’s foot.

Cutting ribbon, string, paper strips to the length of various things (e.g., width of door, height of blackboard).

Constructing “lines” of blocks or other material (representing walls or roads), constructing other “lines” equal

in length to the first one. Rebuilding the “lines” with the same materials but in a different order.

Thomas

Frederick

Jason



Playing “Mr. Here” – hiding a doll and then give directions to find it – “it is near, it is far, too high, too low” etc.

Perimeter Activities:

Cutting strips of materials to fit around head and waist,

Running around the oval.

Walking around shapes on the ground.

Tying string around objects (how much do we need?)

Putting tape around objects.


Directly comparing lengths where both objects can be moved for the comparison: e.g., heights

of children, ribbons, strips of paper, string, pencils, sticks, etc. Also should compare

thicknesses and widths of objects. Introduce language such as shallow and deep, etc.

Comparing lengths where only one of the objects can be moved: e.g., stick with side of

a table, string with height of shelves, child with height of a cupboard, roll of toilet paper

with the distance around the oval, span of a child with width of doorway, etc.

Comparing lengths through the use of an intermediary: e.g., string, paper strips, body

lengths – width of table and width of door, length of blackboard and height of cupboard,

distance around a can and length of pencil, etc.

Getting Close

An interesting idea for a length activity from Burns, M., I hate mathematics book.

Find a group of birds (or animals) – say pigeons. Pick a pigeon. Walk slowly towards it. Try not to make

sudden movements or noises. Try to look inconspicuous. When it flies away, drop something. Keep your

eye on the spot where the pigeon was. Mark the length between where pigeon was and the dropped

object on a piece of string. Do this 10 times. Is there a pattern? What is the average fright distance for

pigeons?

Try the same experiment with people. Talk about something as you approach another person. When do

they begin to look uncomfortable and back off? Mark this distance. Compare the pigeons’ fright distance

with that of people. Try the dame experiment to find the distance at which, speaking normally, you can

be heard and understood by another person. Walk towards this person talking quietly until they motion

you to stop. The person could have his/her back turned.



Comparing different things: e.g., width and thickness, nearness of chair and height of ceiling, perimeter and

length, etc. Ordering objects from shortest to longest/tallest. Categorising objects by near and far, nearby

and away, close and distant, etc.

Perimeter Activities:

Perimeter to length activities:

rolling bicycle wheel one revolution along a wooden plank

wrapping a paper strip around a cylinder/can

opening out a wire rectangle to compare it with a stick, etc.

Perimeter to perimeter activities:

using string or paper as an intermediary,

opening out wire or geostrip shapes and comparing the total lengths of their sides

rolling two cans one revolution each to see which can rolls further, etc.

Drying glasses (an idea from the ‘I hate mathematics’ book)

When you dry a glass with a towel, you dry up the side and around the top rim. Which is

longer, the distance up the side (the height) or the distance around the top (the

circumference of the circle)? Use your towel. Mark off the height of the glass on the

towel. See if this distance will wrap around the top?

Bet the rest of the drying up that a glass can’t be found for which the height is longer than the

circumference. You’re sure to win!

Skeletons

Each member of the group (three pairs to a group) chooses a different coloured ribbon. Working in pairs and

using the ribbon, cut off TWO lengths of ribbon to correspond to perimeter of head (H), perimeter of face (F),

perimeter of left foot (L), perimeter of neck (N), waist (W), total height (T) and arm span (A). Compare the

lengths of ribbon. Can any interesting relationships be found between any of these measures?

Make comparisons between the students: Head & Face, Neck & Waist, Height & Arm span, Head & left foot,

Face & Left foot.

Record which is the larger. Are you a square (Total height = Arm span), a tall rectangle (Total height > Arm

span) or a wide rectangle (Total height < Arm span)?

Use one set of ribbons to prepare a graph showing the variations of Head, Face, Neck, Waist, Total height,

Arm span and Left foot within your group.

Use the second set of ribbons to make 6 skeletons for your group: 2 full size, 2 half size and 2 quarter size. What other measures do you need to do this? What happens to the cross section area and volume of you head when you halve and quarter the perimeters? What age children would correspond to your half size and quarter size children? Check this with actual children of these ages. What do you notice?




Using various body parts to measure length

Using various objects to measure length, e.g., pencils, pencil cases, sheets or paper, blackboard dusters,

Smarties, Cuisenaire rods, blocks, straws, cardboard strips, pegs, paper clips, lengths or dowel, length of

string, etc. Good to have things that are multiples of lengths of other things. Cut out a copy of your foot. Use

this as a non-standard measure.

Using anything that rolls and counting the revolutions turned when the object is “wheeled” along the length,

e.g., bicycle wheel, cotton reel, can, dish, etc.

Perimeter: All the above but going “around the outside” of a shape. The rolling

method is particularly good when the shape is irregular. So is putting string around

the shape and then measuring the string.


Introducing the kilometre, metre, centimetre and millimetre (e.g., constructing the metre out of graph paper

strips and the centimetre out of straws, cutting paper strips to the length of 1 or 2 millimetres, walking a

kilometre). Measuring parts of their bodies, measuring common objects (e.g., cars, door heights,

blackboards, books, sheets of paper, ports, distance from school to home, distance around the oval, etc).

Finding things shorter than, and longer than, 1 metre (m).

Measuring in a variety of situations, e.g., diameters (internal and external), thickness of gaps, depth, etc.

Using a variety of measuring instruments, e.g., rulers, tapes. Using rolling measurers, for example, metre

trundle wheels, or other wheels calibrated to standard units (can get a small trundle wheel for cm).

Perimeter: As above, but going around the outside of a shape. Again it is often useful to use an intermediary

(like string).


Most length activity in this section is in relation to other attributes and appears in Unit 7.4. An important area

of focus is special lengths, e.g., diameter, radius, base, height, etc, and especially perpendicular distance

(shortest distance).

Perimeter: Discovering the formulae for perimeter of a square and rectangle. Draw shapes on graph paper

and record lengths, widths, and perimeters on a table for discussion. Cut out rectangle shape from paper

and cut along diagonal and form 2 triangles to show that the total perimeter is two lengths and two widths.

W 2W

L 2L

Discovering the formula for circumference of a circle using cylinders, string and calculators, or by rolling the



circle out along a page and then seeing how many times the circle by diameter fits into this line.

AREA ACTIVITIES

Stage 1 – Identifying the attribute

Dressing dolls, wrapping packages, wrapping strangely shaped (or hard to wrap) parcels, painting and

colouring in.

Covering desktops with paper, covering play areas with paper, cutting and pasting one shape to cover

another (i.e., a square piece of paper to cover a long thin rectangle of same area.).

Using stamps or paint on hands to cover a space with colour. Use wet hands to cover part of the blackboard.


Directly comparing areas by placing on top of each other, e.g., table on top of table, book on top of chair,

pencil case on top of sheet of paper, etc. Drawing around body (or part of body) and comparing this with

another child’s body (or part).

Indirectly comparing area by covering one instance with paper and then transferring this paper to another

instance. This is very good for instances of differing shape where the paper has to be cut and rejoined to fit

on the other instance. Using other material to do the same thing, e.g., cloth, plastic, etc.

Using dissections, e.g., tangrams (and other shape puzzles), to cover space, and then breaking the

dissections apart and rejoining differently to cover other spaces.


Using a variety of materials as units, e.g., dusters, blocks, cubes, tiles, sheets of paper, hands, etc.

Particularly interesting are chalk-filled dusters on paper, stamps on paper, wet hands on blackboard, etc.

Use geoboards, both square and isometric (triangles).

Using tessellations as non-standard units, e.g., triangles, quadrilaterals, hexagons, Escher type drawings,

etc. Make plastic overlays of these tessellations (e.g., O/H plastic) so can place easily over shapes.

Otherwise cut out shapes and place on top of tessellating grids. Note – Do no be afraid to use non-

tessellating shapes to cover and count.

Using tangrams to cover shapes and assigning unit value to one of the tangram pieces (see page 29 of this

book).


Construct a square cm and a square m. Attempt to experience a hectare and a square km, by walking

around such a space.

Use cm graph paper (preferably in the form of plastic overlays) to measure are of hand,

A4 paper, bag, etc, (anything that is an everyday item for the child). Use centicubes or

MAB units and pack into spaces. Use geoboards (1cm squares) and rubber bands.



Learn the two techniques in this book (‘breaking into parts’ and ‘enclosing with a rectangle”), plus the

technique of finding the area of a triangle by halving the appropriate rectangle. Use these techniques to

calculate the areas of polygons (any type). Calculate the area inside curved shapes by counting the squares

that are all or more than half way inside the shape (or by counting squares all in and squares partly in and

averaging).


Discover the formulae for the area of a rectangle (or a square) by constructing a variety of rectangles on

graph paper (or geoboards) and recording length, width and area (by counting squares) – see pages 106-

107 of this book. The area formulae for triangles and parallelograms can also be discovered in a similar

manner to this.

Relate the area of a parallelogram to the rectangle by cutting and moving one end of the parallelogram (see

page 111 of this book). Relate the area of two triangles to a rectangle using the techniques of page 111 of

this book. In this way the area formulae for triangles and parallelograms can be related to the area formula

for rectangle.

By cutting a circle into sectors and rearranging the sectors to form a rectangle (see below) of dimensions half

the circumference by the radius, the formulae of the area of a circle can be discovered.

Calculate the areas of standard figures by using the formulae. Measure examples of these figures in the

environment (e.g., tennis courts, rooms, etc) and calculate their area.

Unit 3.1.2 Volume and Mass Activities

Focus:

This unit focuses on classroom activities for volume, capacity and mass.

Background:

Volume is the measure of three-dimensional space. Mass is a measure of the inertia of an object – the

amount of effort to move and stop it moving.

Volume activities are usually associated with filling and packing and mass activities with pressure

downwards from gravity.

Activities:

Materials required: Pen, paper, a library or a collection of mathematics education resources materials for the

activities as described below.

VOLUME AND CAPACITY ACTIVITIES

Stage 1 – identifying the attribute

Sand and water play, pouring from one container to another, filling and emptying,

building sandcastles, building with blocks.

Packing things away, filling a box or carton with material, filling a bucket with junk material. Enclosing space,

building a house or a fort, stacking materials.



Blowing up a balloon, making cakes (or any type of cooking), mixing ingredients, acting out or miming big

and small with children’s bodies, enclosing space with children’s bodies. Acting out stories like ‘Three

Billygoats Gruff’, ‘Goldilocks and the Three Bears’, etc...

Immersing objects in water and watching the level rise or the water spill over.

Stage 2 – Comparing and Ordering

In simple cases, directly comparing by placing one object inside the other. Use 3D shape puzzles (eg soma

cubes).

Indirectly comparing and ordering by pouring water, sand, rice macaroni, etc. From one

container to another and seeing which container holds more.

Doing the same thing by immersing objects in water – determining the larger object

either by the height the water rises or the amount that is spilled over (in both cases the

water level must be the same to start). Two interesting ways to o this are to mark the

increase for one object and to see if the other makes the water go higher or lower, or to

fill a bucket to the brim with the first object in it and then to remove this object and to

place in a second to see if there is any spill.

Discuss empty and full, large and small, etc.. Order pictures of objects or pictures of partly filled glasses.

Stage 3 – Non-standard Units

Packing the container with small objects and counting them, e.g. cubes, lollies marbles,

etc.. Using an tessellating three dimensional object in a similar manner.

Pour water etc. Into containers from smaller containers and count how many small

containers full. Use spoons, thimbles, lids, small glasses, egg cups, buckets, etc.

Make a homemade ‘measuring cylinder’, calibrate by regularly spaced lines, from a glass jar

which has a constant diameter. Use this to measure volume of containers, by pouring water

etc. Into the ‘cylinder’ from the containers, and to measure volume of objects, either by immersion and rise in

water level (get the level right to start with) or by measuring the overflow.


Construct a cubic cm, a cubic m and a l (see page 81-82 of this book). Look at L bottles and packages (and

other regular sizes). Calibrate a 1 L milk carton into 1L mL intervals. Calibrate a glass jar similarly. Classify

containers as larger and smaller than 1L.

Relate mL to cubic c L to 10 cm cube and kL to cubic metre.

Use a measuring cylinder. Measure the volume of everyday things such as a cup, a glass, a

child’s fist, etc.. Measure directly or by immersion. Look at the rise in level or collect the

overflow (there are commercial examples of overflow measurers).

Pack containers with cm cubes. Construct rectangular prisms (and cubes) from cm cubes.

Stage 5 – Formulae and Applications

Discover the formula for the volume of a rectangular prism directly by constructing prisms

and recording lengths, widths, heights and volumes or from the area of a rectangle by

constructing prisms of height 1, 2, 3 (and so on) units (see ways A and B on pages 107-108

of the text). Calculate volumes after measuring lengths of sides.



Relate cylinders to prisms to justify the formula for the volume of a cylinder.

Construct a cylinder and a cone of the same height and

diameter and pour cones full of sand into the cylinder to

show that the volume of a cylinder is three times the volume

of a cone.

Hint: Make the cone first – cut the cylinder to the cone’s height.

MASS ACTIVITIES

Stage 1 – Identifying the attribute.

Hefting objects, trying to lift different objects – ‘how much can I lift?’ (be careful here).

Looking at how hard things press down on the hand. Placing things on the end (or

middle) of a stick and seeing how much the stick bends (or bows).

The balance beam – trying different objects on each side, using a see-saw and the

children themselves. Developing the notion of balance and how we can balance

and unbalance. Look at questions such as “how can we get this side to go up?”.

Using a spring (or a rubber band) to see how long different things stretch out the spring when

lifted with it or how much they compress when the spring is placed on top of it.


Comparing two objects by hefting them. Using a beam balance to compare two objects. A homemade

beam balance can be constructed from a wire coat hanger, string and two margarine containers (see below).

A homemade spring balance or rubber band balance can be constructed from a string, a wire hook, a

margarine container and rubber bands (see above). This balance can be used to compare mass by seeing

which objects stretches out the rubber band the furthest. Mass can also be compared with a flotation

measurer (see above). This is a tall container floating in water with sufficient plasticine in the bottom to

prevent it tipping. Objects placed in the container cause it to sink deeper into the water. The object causing

the furthest sinking is the one with the most mass.

Other homemade mass measurers can be made from pieces of wood or plastics or metal as below (rubber

bands can also be used in the RHS measurer). How much the materials bend or stretches determines the

mass of the object.

Stage 3 – Non-standard Units

Use beam balance and determine the number of small objects required to balance

the object being measured. Small objects that can be used include dusters, ball bearings,

marbles, books, MAB units, stones, pencils, etc.

The homemade spring balance can also be placed against a blackboard or a sheet of paper. The container

can be weighted with plasticine to make the rubber bands taut. This position can be marked and called 0.



Then regular intervals can be marked on the blackboard/sheet going down. When objects are placed in the

container, their mass can be read off the position that the rubber band stretching allows the container to

move down to.

A similar mass measurer can be made by marking regular intervals on the side of the flotation

measurer.

Both the spring balance and the flotation measurer can also be used similar to the beam balance.

An object is placed in the container and the position where the container ends up is marked. The

object is removed and smaller objects added until the same position is reached.


Construct and heft a g and a kg. Measure out on a balance material to equal 500g, 250g, etc. Children

measure their own mass and the mass of the teacher.

Use a variety of balances and standard masses to measure a variety of everyday objects. Use beam, spring,

bathroom and kitchen balances.

Calibrate a homemade spring balance and a homemade flotation measurer by adding 100g masses and

marking. Stress that 1ml of water is extremely close to 1g in mass and 1L of water is extremely close to 1kg.

Measure the water that overflows when an object is added to a flotation measurer floating on top of a full

container. The mass of the water overflowing is the same as the mass of the object added. In fact, the

object’s mass is the same as the volume of the overflowing water.

Stage 5 – Formulae and Applications

Mass really requires comparison with other attributes to give rise to many investigations. There is no

formula. Nett and Gross mass are worth investigating (e.g. in contents of a jam jar, etc.)

Teaching Hints:

1. The relation between mass and volume is at the basis of much of the above activity. There will be

more on this in Unit 7.4. Children should be free to explore these relationships.

2. Obviously, volume and capacity involve water and sand play and will need to be especially planned

for. It is a good opportunity to get outside.

Unit 3.1.3 Time, Angle, Money and Temperature Activities

Focus:

This unit focuses on classroom activities for time, angle, money/value and temperature.

Background:

These four attributes are not as straightforward as length, area, volume and mass in the application of the 5

stages. As stated in the overview to the Chapter (page 147), time-telling and money-handling skills have to

be taught separately. There is not the range of activities in each stage.

The stages are applicable to the four attributes and will enrich their teaching if used in planning.

Activities:

Materials required: Pen, paper, a library or a collection of mathematics education resources, materials as

described in the activities.



TIME ACTIVITIES

Time-Telling skill

Turns part turns quarter and a half turns, full turns. Studying the clock face and the two

hands, working with non-geared and geared clocks (see pages 102-103 of this book).

Clockwise and anti-clockwise turns, relating full turn of large hand to movement of small

hand. Relating a full turn to an hour. O’clock, half-past, quarter past and quarter to.

Geared clock activities – discovering relation between small hand and large hand

movements (see the Nelson article, Arithmetic Teacher May 1982).

Focusing on fraction of a full turn – angle wheel, rotagrams, rotascan clock. (see below).

Counting by fives, telling time in five-minute intervals (5 past, 10 past, etc.).

Introducing sixtieths in relation to clock face.

Introducing notion of minute.

Reading time on digital clocks.

24-hour clock.

Relating, eg, 5:42 to 17 minutes to 6.

Worksheets with clock faces, relating drawings of time on face with language and digital time. Relating

different ways to use language to tell time (see page 102 of this book).

Introducing the second, using stopwatches. Applications to sport.

Stage 1 – Identifying the Attribute.

Arranging pictures of events in the order in which they happened hanging pictures with

pegs on a clothes line in this order (with the size of the pictures related to the length of

the event and the closeness of the pictures related to how closely they occur next to

each other in time).

Associating events with the time of the day, eg when do we come to school? Looking

at how cycles repeat themselves, e.g. out of bed, dress, eat, clean teeth, etc.

Experiencing activities that take a short or a long time. Waiting while others do things

(e.g. hop ten times, run around oval, etc.). Noting when things start and finish.

Experiencing a variety of activities which all take the same time. Relating the passing

of time to another activity, e.g. shading in a column or row on a blackboard.

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Directly comparing which activity takes longer by “having a race”, e.g. time for a ball to

stop bouncing and walking across a room, running around building and writing name and

address, etc. Ensure children start both activities at the same time.

Indirectly comparing activities by timing both events with the same technique, ie the amount of

sand to run out of a small hole, the length of a taper candle that is burnt down, the length of the

column on a blackboard that is shaded, the distance a ball rolls down an incline, the depth a

holed container sinks to, etc.

Relating events to a common event, e.g. eating breakfast is longer than a short cartoon on TV but dressing

takes less time, etc.



Labeling strips of paper with events, the length of the strips of paper is determined by the length of time the

event took.


Using a regular action to time events, e.g. counting, pulse, a pendulum’s swing, hopping, the roll

of a ball bearing in a curved groove, etc. (see pages 44-47 of this book).

Calibrating the regular change in something to time events, e.g. a candle burning down is marked

with pins at regular distances (thin taper candles are best), water or sand pouring out a small hole

into another container marked into height intervals, a container marked at intervals down its side

sinking in water, a ball rolling slowly down an incline marked into distance intervals, etc..

Relating time to the movement of the sun. Constructing a sundial. Discuss historical time measurers.

Discussing terms such as midday, morning, afternoon, evening (use worksheets and relate times to

drawings).


Introducing minutes, hours, seconds, clock faces, digital time, 24-hour clock. Determining what children can

do in a minute. Working out the length of a favourite TV program.

Experiencing a minute and other time intervals by doing something for this time. Relating these

times to everyday events like brushing teeth or eating, or school lessons, etc. Turning back and

calling out when 20 seconds is up. Doing this while doing something else, e.g. hopping, holding

breath, etc. Choose actions that may change perception of time.

Using stop watches and clocks, etc. to time events. Calibrating a sun dial in hour e.g. putting a stick in the

ground and marking where shadow points when each hour is up.

Relating daily events to the times they commonly occur (use worksheets and join pictures to times). Working

out a roster (daily or for a special activity). Relating digital time to clock faces.


There are no formulae. Applications are in relation to other attributes, mainly in the context of rate, e.g.

speed, water wasted by a dripping tap, etc.

Many good mathematics process problems are related to clock faces and to timing events.

ANGLE ACTIVITIES


Discussing things that turn, experiencing turning with the children’s bodies, following

the turns of another (“follow the leader”, “simon says”, etc.). Relating the turns to the

directions to which the body points at the starts and ends of the turns. Rotagram and

angle wheel activities.

Turning the pages of a book, turning geostrips in relation to each other. Giving directions to a blindfolded

partner (e.g. “turn right, turn left, turn further, ...”, etc.).


Comparing angles by taking a copy of one angle as a sector of a circle and

placing this copy over the other angle. The copy can be made on tracing paper

and cut out or by using an angle wheel or rotagram (see below).



An angle can also be copied with two hinged strips joined by string a set distance from the hinge. In this

angle copier, the length of the string determines the size of the angle (see above). A compass and a ruler

can also copy an angle by measuring the distance between the two rays (or directions) with the ruler along

an arc a fixed radius from the vertex (arc is drawn by compass held at a fixed radius).

Constructing a right angle measurer by folding paper and comparing with angles to classify them as acute or

obtuse (see below for fold).


Measuring extent of turn by how many small sectors will fit in the angle. Can also get

approximate measures by how many of an object can be fitted into the angle at a fixed

distance from the vertex, e.g. fingers, pencils, etc. A rotagram, angle wheel or a home-

made protractor can be calibrated with regular intervals along an arc and the angle read

from these intervals (see below).


Constructing a sector equal to 1 or 2 degrees. Experiencing the protractor. Measuring common angles.

Using an inclinometer to measure incline (see below for how to make a home-made

one from ruler, protractor, string and mass).


Tearing off the corners of polygons to determine the interior angle sum rules for these

polygons. Triangulating polygons into triangles for the same purpose (see Geometry text).

“Walking” around regular polygons to find relation of number of sides to the exterior angle

(the amount that has to be turned in, e.g. LOGO).

Relating angles in regular polygons, relating interior and exterior angles in triangles, relating

angles in circles (particularly those from diameters), relating angles in parallel line

situations, relating angles between diagonals to the various quadrilaterals, relating angles

to tessellation, relating angles in line and rotational symmetry, etc. (all this is in the

Geometry text).

3. Read the following activities for money/value. In the space left on the RHS of the descriptions, write in

extra activities. Look through mathematics education resources in your library for good ideas.

MONEY/VALUE ACTIVITIES

Money-handling skill

Recognition of coins – coin rubbings, drawing coins, coin posters, etc.

Value of coins – relating coins to each other, recognising what can be bought for certain

coins, adding and subtracting coins and making change.

Setting up a “shop” – things to buy and sell, play money, etc. Computation and word problems in money

situations.




Discussing what things are the most valuable for them (e.g. “what one thing would you take to be marooned

with, ...”, etc.). Discussing why they value it. How they would show it had value. Choosing amongst

alternatives.

Discussing things that have little value. Pictures of expensive and cheap things.


Subjectively comparing objects to see which one is valued more highly. In directly comparing by

relating to a common object, e.g. I like my toy more than this pen but the trip to get hamburgers

is better than the toy, etc.

Bartering – discussing how many of one object is a fair trade for another. Organising the class into groups all

with a different product and discussing how the tribes would barter one thing for another (introduces supply

and demand).


Discussing the use of a common or valued object as a medium of exchange, e.g. Cowrie shells, oxen or pigs

or horses or other animals, marbles, etc. Discuss historical development of money.

Discuss the talent, an iron ring that had the mass of the gold needed to buy an ox – became a basis for

money and mass.


See money handling skills above.

Play “shop”, determine everyday items that can be bought for the coin/note under

discussion or multiples of that coin/note.

If possible, involve children in actual buying and selling, e.g. tuckshop activities –

organising lunches and lunch money, buying, cooking and making and selling food on days

the tuckshop is closed, organising an outing and determining the money needed, etc.


Other than early work on interest and percentage there is little formulae here. There is a huge supply of

money problems both routine and process.

TEMPERATURE ACTIVITIES


Experiencing a variety of temperatures. Describing different types of days (e.g. a summer

scorcher or a winter frost, etc.). Heating ice until it boils. Pictures of hot and cold. Reading

stories involving temperature (e.g. Goldilocks and the Three Bears).

Feeling the temperature of your head after your hand has been in hot water and then after

your hand has been in cold water.


Comparing temperatures directly with the hand, moving hand from one instance to another, the

hand will feel the change. Relating temperatures by comparison with or to known things, e.g.

that one is hotter than my head feels yet this one is not, etc. Fill cups with water at different

temperatures and let children order the cups from warmest to coldest.

Can indirectly relate temperatures by what they do to other things, e.g. the paraffin is runny in



that situation but remains solid here, etc.

Constructing a simple temperature measurer with a thin tube and coloured water in a small

container. Discovering how the coloured water goes up the tube as the temperature rises.

Discuss hot days and cold days, order pictures that show hot and cold. Discuss summer

and winter, what is common to wear, what is commonly eaten and where is the common

vacation area. Keep a weather chart, drawing and describing the weather on each day.

Relate this work to other subjects, e.g. Science (effect of heat and cold on humans and

animals, also on food and other materials), Language (talk about feelings and observations

concerning different types of weather and its effects on them), Environmental studies, (how

changes of temperature affect things such as butter, care of pets in hot and cold

conditions) and Health (survival in hot and cold situations).

“How does it feel!”

1. Have students place their hands on their cheeks to feel how warm or cool their hands are, then have the

children rub their hands together briskly for about 30 seconds, and put them against their cheeks again. Ask

the children if their hands are warmer or cooler after they rubbed them together?

2. Arbitrarily label several spots in the room which have different temperatures for the children to touch; for

example, window glass, metal shelf, spot in the sunlight, spot in the shade, and so on. Let the pupils try to

determine which spot has more heat and which spot has less heat. Let the students make statements such

as, “The metal shelf has more heat than the window glass.” Let the children attempt to order the spots by the

amount of heat they feel. Point out to the children that the things that have more heat than their hands feel

warm or hot to them and vice versa.

3. Obtain two glasses of water from the tap, and let the children feel them to see that they are both cool. Set

one in the shade and one in the sun. Thirty minutes later, let the children feel them again. Discuss why one

glass now feels warm, whereas the other is still cool.

4. Let two pupil “judges” stand in front of the room. Have 6 or so pupils file past and lay their hands on the

“judges” cheeks. The “judges” try to decide who has the coldest/warmest hands.


The only possibility is to calibrate the measurer above by regular distance intervals along the length of the

tube. This could be used to order the cups of different temperature water.


Learn to use a thermometer. Familiarise the children with how temperature change

causes changes to the height of mercury in the thermometer. Let children discover

this movement of the mercury.

Get a feel for the degree by feeling water that has been heated 5 degrees, say, in comparison with water

from the tap.

Measure the temperature of common things, e.g. ice, boiling water (be careful!), body temperature, etc. Use

a thermometer to find the warmest and coldest spots in the classroom.


There are no formulae, but some interesting investigations can be held around boiling and freezing points,

e.g. why does salty water not freeze as readily as ordinary water? etc.



Teaching Hints:

Many excellent suggestions for measurement activities are to be found in primary subjects other than

science. For example, a social studies module on Eskimos would have many opportunities for relevant work

on temperature.

Unit 3.1.4 Activities for relating attributes and workstations

Focus:

This unit completes the survey of classroom activities for measurement by focusing on activities that relate

attributes. It also gives examples of investigatory activities that can be the basis of rotating workstations and

examples of measurement experiences for the early childhood years.

Background:

Many of the most interesting measurement activities with the greatest problem-solving capacity involve more

than one attribute. In fact, they are based on the interrelation between two attributes. For example, mass and

volume, two attributes that are commonly confused, interrelate as density with many interesting conclusions.

One particularly enticing investigation can be studied with a displacement experiment. If an object is

immersed in a full bucket of water, the resulting overflow will equal the volume of the object. If, on the other

hand, the object is placed in a container floating on top of a full bucket of water, the resulting overflow will

equal the mass of the object (if it does not sink).

These types of activities often involve a longer time than the normal class and an intricate equipment set-up.

It is difficult to have multiple copies of these set-ups. One way to overcome this is to organise different

experiments and materials in different positions and rotate the children, in groups, through each experiment.

This teaching approach is called “rotating workstations”. Of course, such workstations can be organised for

individual attributes as well as situations where more than one attribute is involved.

Activities:

Materials required: Pen, paper, a library or other collection of mathematics education resources, materials as

required for the activities below.

ACTIVITIES FOR RELATING ATTRIBUTES

Length/Perimeter Area:

Cutting up “fat” rectangles of paper to cover long thing rectangles.

Drawing shapes with a fixed perimeter and determining which

shape has the largest area for a given perimeter (it’s the circle). (Restricting this to

rectangles gives the square as the largest area for the smallest perimeter – this is

why cheap houses are square.)

Starting with a fixed number of blocks and making a variety of shapes of the same area and comparing their

perimeters.

Length Volume:

Pouring from long thin containers into short fat containers. Studying packaging in

supermarkets. Comparing litre soft drink bottles with MAB blocks (both 1 L).

Length Time:



Looking at speed in stories such as the Tortoise and the Hare. Measuring how far can walk or run in 1

minute and using this to calculate how long to walk or run 1km.

Area Volume:

Comparing surface area with volume and also area of base with volume, particularly in relation to height, e.g.

halve the length and breadth of the base, what happens to the height if the volume wishes to stay the same?

Roll ups (an idea from the ‘The I hate mathematics’ book)

Obtain 2 pieces of A4 paper, tape, extra paper and some beans. Roll the two pieces of paper into cylinders,

one “longways” and one “shortways” (see below). Add a bottom to each container and stick down. Use the

beans to determine which container holds the most.

What is the better container, the short fat one or the tall skinny

one?

Look after the baby

Make a shape from 5 square counters. Note the area and perimeter. Make the shape again but double

original scale, and again, but triple. Note the areas and perimeters for all three. Record this information on a

table. What is the pattern in the relationship between area and perimeter as the size increases?

Repeat the above for a 9-counter square, doubling and tripling the length and width. What pattern is

emerging in the perimeter area relationship?

Start with 3 cubes and make any 3D shape. Double and triple this in scale. Calculate volume and surface

area. What is the effect on surface are of doubling and tripling lengths of sides? What is the effect on

volume?

Make cubes of side 1, 2, 3, 4, 5 and 6. Calculate area of base, volume and surface area for these cubes. Put

these on a table. What pattern emerges? What happens to the surface area to volume ratio? What important

consequence does this have for babies?

Mass Volume:

Hefting large amounts of low-density material (e.g. pillows) and small amounts of

high density material (e.g. lead weights). Balancing dusters and bricks. Looking at

pictures and discussing big and heavy, small and light, small and heavy, etc.

Investigating density. Using the flotation mass measurer (see Unit 7.2) to determine

that the water displaced by a floating object equals the mass of the object and using a

displacement volume measurer to determine that the water displaced by an immersed

object represents the volume of the object.

Money Volume:

Discussing what happens when we add more to the pile being valued. Valuing such things as large and

small blocks of chocolate, diamonds versus pillows, etc.

WORKSTATION ACTIVITIES

A. Comparing and ordering workstations

Station 1: Length

Materials – string, ball, pen and paper.



Use string to determine which is the larger

The distance around the ball OR the length of the blackboard compass (closed)?

The height of glass cupboard OR the length of blackboard?

The distance up and down the six flights of stairs outside OR the distance from the maths building to

the stairs in front of the refectory?

Station 2: Area

Materials – cardboard carton, paper, scissors, tape, pen and paper.

What has the larger area?

The bottom of the carton OR the base of the chair?

Area covered by the hat OR two A4 sheets?

Station 3: Volume

Materials – sand, containers, pen and paper.

Place the containers in order from largest to smallest volume.

Station 4: Mass

Materials – coat hanger, string, margarine containers, scissors, objects, pen and paper.

Construct a balance with the coat hanger, string and margarine containers. Use this balance to place the

objects in order from largest to smallest mass.

Station 5: Time

Materials – ball, pen and paper.

1) Which takes longer

Walking across the room OR tying your shoe laces?

Time taken for a bouncing ball to stop OR writing the heading of this handout?

2) Make up 5 activities to time (be creative) and place in order.

Station 6: Angle

Materials – paper, scissors, cars, pen and paper.

By cutting paper copies of the angles,

Determine which can form the sharper angle, your elbow OR your knee.

Place cars in order from largest to smallest by the angle their front door opens.

Station 7: Money/Value

Materials – Pen and paper.

1) What box of materials in this room would you most value in your classroom next year and why?



2) Choose six important necessities of a student’s life other than money (e.g. shelter, food, etc.) and

order them in terms of importance to you. Is this the same order that you would spend money on

them?

Station 8: Temperature

Materials – building, pictures of hot and cold, pen and paper.

1) Order the pictures from coldest to hottest.

2) Find five objects of different temperatures in this building and order from hottest to coldest.

B. Non-standard Unit Workstations

1. Body-length

Materials – blackboard, body lengths, pen and paper.

Directions – use the following body-lengths to measure the length of the blackboard estimate first and

complete the following table.

Estimate Measure

Hand Spans

Cubits

Foot Lengths

2. Cardboard-area

Materials – large piece of cardboard blackboard duster, A4 pages, pen and paper.

Directions – use the following materials to measure the area of the cardboard, estimate first and complete

the following table.

Estimate Measure

Hands

Blackboard Duster

A4 Pages

3. Volume measurer

Materials – coffee jar marked with felt pen divisions, water, three containers, pen and paper.

Directions – use the marked coffee jar to measure the volume of the three containers, estimate first and

complete the following table.



Estimate Measure

Container 1

Container 2

Container 3

4. Mass measurer

Materials – margarine container connected to a rubber band hung against a page with divisions, three

objects, pen and paper.

Directions – use the mass measurer to measure the mass of the three objects, estimate first and complete

the following table.

Estimate Measure

Object 1

Object 2

Object 3

5. Tangrams and area

Materials – tangram sets, worksheets pages 30 – 32, pen and paper.

Directions – use the tangrams to complete cards C, D, E, F, G and H, estimate first and complete the

following table. See Comparison and order tangram cards earlier in the book.

Estimate Measure

Card C

Card D

Card E

Card F

Card G

Card H

6. Pendulums and time

Materials – two pendulums of different string length, pen and paper.

Directions – use the two pendulums to time three activities, estimate first and complete the following table.

Estimate for

Pendulum 1

Measure for

Pendulum 1

Estimate for

Pendulum 2

Measure for

Pendulum 2

Activity 1

Activity 2

Activity 3

C. Measurement Investigations



Most of these investigations are taken from LeBlanc, J. F. et al (1973). Measurement (A MMP book)

Bloomington: Indiana University, acknowledged at the beginning of this book.

A. How many of your hands to cover the surface of

your body? Is there a relationship between hand

area and body area? What is this measure used

for? (Hint – its medical).

B. Drop a ball from different heights and measure

the height of the bounce. What is the relationship, if

any, between these two quantities? Does the

relationship vary with the ball? Repeat the task with

a ball of different size or elasticity. C. The bag of scraps is left over from making a

garment from material which is 150cm wide. Given

the length of material (written on the bag), how

much does the garment weight? (not including zips

etc.)

D. What is the effect on the volume of a cube if

you double the length of the sides? How about if

you triple the length of each side? What is the

effect on the volume of a sphere if you double or

triple its radius?

E. What relationships, if any, exist between

volume and surface area as size increases?

Record results for cubes of sides 1, 2, 3, 4, 5

and 6. What implication does this have for

babies? F. Find 10 measures which might answer the question, “How big are you?” Collect the data for each

measure from the members of your group. Do any two measures seem to correlate (i.e., a person who

has a large measurement tends to have a large measurement in the other)? Plot the data for two

correlated measures on graph paper. (You might want to get these measures from other members of

the class in order to have more data.) Does your graph have an interesting shape?

G Make a cylinder and a cone which have the same vertical height

and the same circular base. What is the relationship between the

volume of the cylinder and the volume of the cone? Check your

answer by constructing another cylinder-cone pair. Do you know a

formula for the volume of a cylinder? If so, what would be the

formula for the volume of a cone?

H. Find the surface area of the

tennis ball.

I. Find the volume and weight of water wasted in 24 hours by a dripping tap.

J. Determine the dimensions of the rectangle of perimeter 36cm that has the greatest area. Use your string and graph paper to experiment. If you are not restricted to rectangles, what shape of perimeter 36cm would you say has the greatest area?

K. Make two cylinders from A4 paper, one rolled lengthwise, the other oppositely. Which holds the most volume? Why? What does this indicate for the surface area/volume relationship?

L. Obtain a bucket of water and a deep container. Place plasticine

in container until it floats without tipping. Use this flotation measure

and a variety of objects to investigate how mass and volume vary.

Check that the amount of water that an object displaces on its own

is its volume, while the amount it displaces when in the flotation

measure is its mass.



CHAPTER 3.2: TEACHING MEASUREMENT

The teaching of measurement requires the active investigation of measurement attributes in situations that

allow the discovery of concepts and formulae and the practice of skills, techniques and standard unit notation

and conversion rates. It also requires taking into account development and maturation (conservation) and

ensuring that students acquire all the understandings in the five stages of measurement instruction.

Identifying the attribute

Comparing and ordering

Non-standard units

Standard units

Formulae

In this chapter, therefore, Unit 3.2.1 looks at conservation and developmental levels, Unit 3.2.2 looks at

sequencing in measurement (writing measurement activities and moving through the five stages) and Unit

3.2.3 looks at diagnosis and remediation (using anecdotes to look for error patterns).

Unit 3.2.1 Conservation and developmental levels

Focus

A young child’s observed performance of many measurement tasks appears to be different to adults’

performance, particularly in how they seem to understand the tasks. This observed difference has lead

psychologists and educators from Piaget onwards to hypothesise the existence of qualitatively different

levels through which children develop. For measurement, the important problem is that of conservation,

realising when the quantity of an attribute does or does not change during a measurement process. This unit

focuses on conservation and developmental levels and how they may affect the teaching of measurement.

Background

The Swiss psychologist Jean Piaget, in his study of how children develop their thinking and grow in their

ability to learn and handle new information has placed warning flags over certain aspects of the

measurement process. When performing measurement tasks, there often comes a time when something has

to be done to the object being measured (e.g., the object is moved or containers are changed). The change

does not affect the quantity of the attribute being measured. We say that the attribute is conserved. In many

basic measurement situations, children have been observed to behave as if they do not believe the quantity

has been conserved.

The development of conservation is, in this theory, a characteristic of the attainment of certain

developmental levels that children must pass through as they move to adult thinking. Young children, not yet

to these levels will not be able to conserve, nor, in some theories, be capable of learning how to conserve.

This can lead to measurement difficulties as the following examples of non conservation show.

(1) Length

A child can be shown two rods of the same length lined up so that it is easily seen that this is so, e.g., …

The child will agree they are the same length.

But if one of the rods is moved as below ….



… the child will now say that the lower rod is longer even though the rods have not changed in length. This

lack if conservation of length is particularly noticeable when a child is given that task of building a tower of

blocks on a lower table to the same height as an already built tower on another table, e.g.,

The child will attempt to build the second tower so that its top matches the top of the existing tower, not so

that its length matches, e.g.,

(2) Area

A child can be show a picture of a paddock with houses (X) in it as below …

The houses can be moved as below …

The child will now say there is less grass in the paddock even though there has been no change in the size

of the field or the number of houses.

(3) Volume

A child can be asked to pour water into a short wide container as on the right. The child can then be directed

to pour that water into a tall thin container as on the right. The child will say that the tall container has more

water regardless of the fact that he/she has just poured the water from the short to the long container.



(4) Mass

A child can make a lump of plasticine to balance a 500 g mass as below …

The child can then be directed to roll the lump of plasticine out into a long “sausage”. The child will now

believe this plasticine has more mass (and also more volume) regardless of the fact that no plasticine has

been added or taken from the lump.

Children are said to attain conservation when they are capable of reversing in their minds the change made

on the object being measured. For example, the child mentally pours back the water from the tall thin

container into the short wide container and, in doing so, comprehends that the volume is unchanged.

Modern theories of developmental levels are based on working memory capacity. It is felt that as children

mature they gain the ability to hold more items of information in working memory. The reversals necessary

for conservation require a capacity, so the theories go, that is not available in young children.

Developmental levels are a contentious issue. Some researchers argue that they are an artifact of the tasks

given to children and the language used in the questions asked of the children. Others argue that, rather

then representing fixed biological stages that children move through, they are only crude indicators of the

sequence through which knowledge is acquired. Whether levels are biologically bases or instructionally

based is an open debate. Some educators argue that a child must be ready in terms of biological maturation

before something can be taught, while others believe that with appropriate instruction any child is ready to

learn anything.

It seems that much of the evidence for levels has assumed a global view of concepts, that acquisition of,

say, number is a single, all or nothing, ability. Modern research is showing that many of these so-called

global concepts are made up of sequences of sub concepts that children can be shown to slowly acquire as

a result of instruction. It also seems that much of the evidence for levels has underestimated the power of

knowledge of context in solving tasks. Modern research is focusing on domain-specific knowledge as having

a strong effect on thinking. Children given tasks in a familiar context show remarkable abilities to confound

the levels has inspired a plethora of powerful modern theories of cognitive development by such researchers

as Pascal-Leonie, Case, Fischer, Collis and Biggs and Halford.

Instructionally, there has been a turning away from levels as a basis for curriculum. The three main reasons

for this are the difficulties in evaluating and determining children’s levels, the existence of many situations

where children operate at different levels in different topics and the ability of children to use task specific

thinking to solve problems amenable, it seemed, only to thinking from a higher development level.

With the literature inconclusive, the best approach for a teacher is to be sensitive to the children and to

realise that: (a) there is prerequisite understanding before a child is ready to learn some things, and (b) there

may be good reasons for so-called wrong answers and for slowness in mastering some activities.



Activities

Materials required: Pen, paper.

1. Read the background to this unit, particularly the four examples of non-conservation. Answer the

following questions.

(a) What difficulties would children who do not conserve length have with a ruler?

(b) What sense would children who do not conserve volume make of a measuring cup?

(c) A child chose the taller container when asked the question “which holds more?’ what could be

the problem with the word ‘more’ here? How could this question be rephrased to ensure the

child’s answer reflected the intended idea of ‘more’?

(d) What difficulties could children have who fail to conserve as in examples (b) and (d)?

(e) Do you think that a child would perform differently in these examples under normal conditions

then under the test conditions that most of these tasks are given to children?

2. There are many situations where adults have difficulty with conservation. For example:

different volume bottles having different shapes;

taller cans having smaller diameters;

1 L soft drink bottles having the same volume as a MAB block; and

a special of 100 sheets for 50c when 500 sheets is normally $2.

(a) Determine which of the following conserves the stated attribute.

OBJECT ATTRIBUTE CHANGE CONSERVED?

Quantity of liquid in a glass

Volume Pour into a measuring cup

Yes

Quantity of liquid in a glass

Volume Freeze the cup of water No

Two sticks Relative length Change positions

Quantity of liquid Temperature Turn on an intense light

Rectangle 10cm x 50cm Area Change to parallelogram of side length 10 x 50

Rectangle Area Cut into triangles

Ingredients of a cake Mass Cook

Ingredients of a cake Volume Mix

$5 note Mass Exchange for 5 x $1 coins

$5 note Monetary value Exchange for 5 x $1 coins

Slum dweller Self-concept Give a job

(b) How important was experience in answering these questions? Where did you learn the answers

to them? At school? Would a young child have been able to answer them? Why?



(c) What effect would it have on one’s ability to measure the attribute in question if you did not know

whether the change was conserved or not? In particular, discuss any questions you got wrong!

(d) What does your attempts at these questions tell you about children who may fail to correctly

answer conservation tasks?

Teaching hints

Take opportunities to trial some of the conservation activities with young children. Try hard not to lead the

children.

If you are in doubt about readiness, then run measurement activities which are exploratory in nature, i.e.,

activities that are open-ended and that do not have a single and point or correct answer which has to be

found. The children simply undertake measurement experiences and describe what they found and what

they think it means.

Do not mark children right/wrong (or demand the “correct answer”) when undertaking measurement activities

where conservation may be a problem. Simply accept whatever children find important and think is correct.

Unit 3.2.2 Sequencing measurement activities

Focus

In the Overview to this book, it was recommended that measurement be developed through five stages:

identifying the attribute; comparing and ordering; non-standard units; standard units; and formulae. This unit

focuses on the use of these five stages to develop teaching sequences of measurement activities for the

primary classroom.

Background

Planning of instruction in schools in both short term and long term. Individual lessons and sequences of

lessons across weeks have to be prepared. Plans have also to be developed for the seven years the

children will be in the primary school. It is important that this planning is done within a framework of how

instruction will fit into children’s long-term experience with measurement. It is also important that each lesson

is part of a short-term sequence to accomplish a certain objective. The material in the Overview of this book

has focused on the important aspects of planning instruction. It should be read again at this point. It is crucial

to not that for whatever planning occurs, implementation of this planning in the classroom must be tempered

by the reactions of the children. If children have difficulty, then lessons/activities must be repeated with

variation until these difficulties disappear. If children find the work easy/boring, then new work can be moved

to at a faster rate than planned for. Planning must prepare the teacher for flexibility in instruction.

Activities

Materials required: Pen, paper, materials as listed for the length cards, scissors.

1. Sequence the following STEPS IN DEVELOPING LENGTH IN THE PRIMARY SCHOOL

A. Introducing the metre.

B. Discussing lengths measured in paces and determining the need for a standard.

C. Investigating relationships between perimeter and area.

D. Comparing two lengths of the same material.

E. Experiences with small units (e.g., mm).

F. Working with Cuisenaire rods to determine that small units give large numbers and vice versa.

G. Connecting diameter and the circumference of a circle.

H. Distances on a globe of the earth/distances in space.

I. Measuring the length around the table and the tennis court.

J. Discovering the formula for the perimeter of a rectangle.



K. Recording measurements in decimal notation.

L. Developing accuracy in length measurement (to nearest mm).

M. Determining what length means.

N. Classroom experiences with a common length unit determined by the class.

O. Finding the perimeter of a collection of objects.

P. Determining which of a collection of objects would be the most appropriate unit for a length measurement task.

Q. Measuring the school with hand spans, cubits, paces, etc.

R. Estimating distances in m and cm.

S. Ordering a collection of objects by length.

T. Experiences with larger units (km).

U. Comparing the circumference of a can with the length of a feather.

V. Understanding all the different words for length (e.g., wide, short, etc).

W. Metric units, conversion rates and decimal numeration.

X. Introducing the centimetre.

Copy and cut out each step. In your groups sequence the steps in the order across the years of primary school they

should be given to children. Give reasons for your sequence.

Is the sequence exhaustive? Does it follow the five stages recommended in the Overview to this book? Brainstorm

any extra steps that should be added to the list!

Do the steps follow a sequence of greater sophistication? Of progressively greater dexterity?

Briefly describe a classroom activity that might undertaken with children for each step.

Prepare a sequence of steps of a similar form for the measurement topic Area!

2. Gather the required material and complete the length work cards in the SAMPLE WORKCARDS ON LENGTH at

the end of this unit. Answer the following questions concerning these cards.

List the objectives of each card! Do the cards follow the sequence of stages recommended in the Overview to this

book? Are there any missing steps?

Why not start at Card 4?

Are there advantages in writing tasks on separate cards as these are? Here are two: Separate cards are more

flexible – extra cards can be inserted for struggling students and cards can be omitted for talented students; and

separate cards give the teacher more freedom – they can be given to a small group of children, to one child, or to

the whole class / the teacher does not have to give continuous directions and therefore is free to help those in

need. Can you think of more?

Was there enough activity in each card? Would you do any cards differently and why? Would children have difficulty

with any cards?

Can you find a card that exemplifies conservation of length?

Choose a measurement topic other than length. Write a series of cards similar to those in 3 above which would

develop the central activities of that topic through the seven years of primary.

Teaching hints

Be a reactive teacher. It is necessary to plan. Find the knowledge level your children have at present in the topic.

Determine where you would like them to be. Prepare a plan that takes the children directly to your end-point as below.



But implement your plan (i.e,. teach) so that you react to the children’s reactions to your teaching by turning

away from the end-point wherever necessary to help the children, using the end-point only as a beacon to

turn towards after the children’s difficulties have been cleared. See below.

When writing work-cards, it is very easy to use language difficult for children to follow. Have selected

children read your cards for sense before you hand them out to the class in general.

Always keep in mind the five stages in the development of measurement, the various steps within each of

these stages and the cycle of activities: development; consolidation; and application. Base your instructional

planning on these.

Unit 3.2.3 Diagnosis and remediation

Focus

In the previous unit, we stressed the need to base instruction on the behaviour of the children. In this unit, we

will focus on how to evaluate children’s performance, to diagnose the underlying cause of any difficulty and

to prepare appropriate remedial activities.

Background

When children have difficulty with a mathematics problem or procedure, teachers should not limit their

assistance to the actual point of difficulty. Giving a rule to get the answer and directing the children to

memorise and practice this rule will not help the children to understand the concept or process behind the

procedure and can have a deleterious effect on attitude.

Teachers should treat a child having difficulty as a sign that understanding may be lacking. The child must

be probed to discover what he/she knows and, hence, what requires reteaching.

The steps are as follows:

(1) observe the children and evaluate their work for any difficulties;

(2) when a difficulty is found, analyse it for any pattern that may indicate the cause of the error / discuss

the difficulty with the child (listen to the child);

(3) list the possible causes of the error and check these with the child, endeavouring to find what the child

knows more than what is not known; and

(4) reteach the missing knowledge starting from what the child knows using the same procedures as if

teaching for the first time.

The basis for determining cause has to be the particular knowledge of concepts, processes and

measurement techniques in each topic area, but also the existence of useable knowledge in each of the five

stages of the measurement process: attribute, ordering, non-standard units, standard units and formulae.



We should also not that evaluating performance in measurement is in some ways more difficult than in

number. Correct and incorrect answers may not be the best indicator or satisfactory performance and

difficulties. In measurement tasks it is possible for more than one solution to be appropriate. It is also

possible to arrive at a satisfactory conclusion to a task and yet exhibit unsatisfactory techniques and

understandings. Likewise an incorrect answer may be the result of one careless measure in an otherwise

flawless performance.

Activities


1. Read the MEASUREMENT ANECDOTES at the end of this unit. For each anecdote, briefly answer

the questions accompanying it. Note: The anecdotes are based on those from the MMP Measurement

book referred to in the acknowledgement of this book.

Teaching hints

Base your evaluation of children’s measurement performance on observations of them in measurement

tasks. In this way, you will be able to determine any difficulties in measurement techniques, use of materials,

problem solving skills, group co-operation and number and space skills as well as the concepts and

processes of measurement.

Look for difficulties as you teach. The best teaching opportunities arise from the statements and questions of

children, particularly those that seem to show that the children is on a different track to you. Do not

categorise the statements as wrong and move on, but probe the children to see the reasons for the

statements. Use it as an opportunity to draw out the concepts or procedures that are the basis of the

statements. Do not be afraid to abandon the planned lesson to follow the questions of your children.

As we have said in 3 above, the most effective teaching action is what you do in immediate response to a

child’s question or statement. To make this response appropriately, you need a good model in your mind of

what complete measurement knowledge is and a belief that all children act from what they think is correct.

The five stages, plus the general objectives, in the Overview at the beginning of this book provide a

recommended model of complete measurement knowledge for the primary school.

Unit 3.2.3 Diagnosis and remediation

Focus

In the previous unit, we stressed the need to base instruction on the behaviour of the children. In this unit, we

will focus on how to evaluate children’s performance, to diagnose the underlying cause of any difficulty and

to prepare appropriate remedial activities.

Background

When children have difficulty with a mathematics problem or procedure, teachers should not limit their

assistance to the actual point of difficulty. Giving a rule to get the answer and directing the children to

memorise and practice this rule will not help the children to understand the concept or process behind the

procedure and can have a deleterious effect on attitude.

Teachers should treat a child having difficulty as a sign that understanding may be lacking. The child must

be probed to discover what he/she knows and, hence, what requires reteaching.

The steps are as follows:

(1) observe the children and evaluate their work for any difficulties;



(2) when a difficulty is found, analyse it for any pattern that may indicate the cause of the error / discuss

the difficulty with the child (listen to the child);

(3) list the possible causes of the error and check these with the child, endeavouring to find what the child

knows more than what is not known; and

(4) reteach the missing knowledge starting from what the child knows using the same procedures as if

teaching for the first time.

The basis for determining cause has to be the particular knowledge of concepts, processes and

measurement techniques in each topic area, but also the existence of useable knowledge in each of the five

stages of the measurement process: attribute, ordering, non-standard units, standard units and formulae. We

should also not that evaluating performance in measurement is in some ways more difficult than in number.

Correct and incorrect answers may not be the best indicator or satisfactory performance and difficulties. In

measurement tasks it is possible for more than one solution to be appropriate. It is also possible to arrive at

a satisfactory conclusion to a task and yet exhibit unsatisfactory techniques and understandings. Likewise an

incorrect answer may be the result of one careless measure in an otherwise flawless performance.

Activities


1. Read the MEASUREMENT ANECDOTES at the end of this unit. For each anecdote, briefly answer

the questions accompanying it. Note: The anecdotes are based on those from the MMP Measurement

book referred to in the acknowledgement of this book.

Teaching hints

Base your evaluation of children’s measurement performance on observations of them in measurement

tasks. In this way, you will be able to determine any difficulties in measurement techniques, use of materials,

problem solving skills, group co-operation and number and space skills as well as the concepts and

processes of measurement.

Look for difficulties as you teach. The best teaching opportunities arise from the statements and questions of

children, particularly those that seem to show that the children is on a different track to you. Do not

categorise the statements as wrong and move on, but probe the children to see the reasons for the

statements. Use it as an opportunity to draw out the concepts or procedures that are the basis of the

statements. Do not be afraid to abandon the planned lesson to follow the questions of your children.

As we have said in 3 above, the most effective teaching action is what you do in immediate response to a

child’s question or statement. To make this response appropriately, you need a good model in your mind of

what complete measurement knowledge is and a belief that all children act from what they think is correct.

The five stages, plus the general objectives, in the Overview at the beginning of this book provide a

recommended model of complete measurement knowledge for the primary school.



SAMPLE WORK CARDS ON LENGTH

These work cards come from the MMP Measurement book referred to in the acknowledgements of this book.

Copyright ended on this material in December 1981.

The cards show a sequence of activities from comparing and ordering to standard units.

Children ready for the later cards (the introduction of the metre) should be allowed to do the early cards even

though they will be easy. The concern of the teacher should be on the methods they use in completing the

cards. If children have difficulty with the early cards, they should be provided with some activities at this level

before proceeding.

Above each card, there is a list of materials for that card. The teachers would supply these materials. They

are listed for the convenience of the reader. Children would not be given the list of materials.

3 pieces of ribbon, red, blue, yellow, in that order of size.

3 pieces of ribbon, blue, red, green, in that order in size.

A feather which is shorter but fatter than a pencil, a piece of string longer than the pencil, a can which has a

perimeter longer than the string.



A can, 3 pieces of string, one shorter than the perimeter of the can, one longer and one the same size, a line

drawn on a piece of card so that it is equal in length to a multiple of “can” perimeters. A piece of blue string

which is twice the length of the line.

Table, chair, and book.

A pencil, a Cuisenaire rod or similar wood or plastic “brick” (about 5 cm long), collection of objects with

straight edges, e.g., Books of different sizes, desk, pieces of card, ribbon, drinking straw.



A stick, unmarked and about 80 cm long.

Metre rule, not marked in centimetres.

Metre trundle wheel.



MEASURMENT ANECDOTES

A. Jane said “I can’t tell whether the table is longer then the cupboard; I don’t have a ruler!”

What is the misconception that Jane has? What is the cause of this misconception? (i.e., what experiences are

lacking from her background?)

What would you do if Jane said this to you when you were teaching?

What sequence of experiences would ensure that this didn’t happen to children in your class?

B. Fred could not work out what to do. Frank had told him that the plane they had visited was 35 paces

long, but what length of string was equal to that?

(1) How could Fred work out how much string to cut? (Can you think of more then one way?)

(2) What experiences seem to be lacking in Fred’s background? What would you do if you were

teaching Fred?

(3) How could you use this to teach the need for a standard?

C. Anne was dismayed. When comparing the two pieces of ribbon, she found that they were not the

same length, yet they were both marked 25 cm!

Is it possible for both Anne and the markings to be correct?

What experiences does Anne lack that would account for her dismay here? (What is the cause of the dismay?)

What might you do with Anne to give her insight into the situation?

How would you use this situation to teach tolerance for error?

D. Henry did not know what to do! His teacher had asked him to measure the school’s fence and had

only given him a 15 cm ruler.

How could you help Henry?

What lack of experiences could be the cause of Henry’s lack of creativeness and flexibility here?

What would you do to promote measurement flexibility in your class so that they did not have Henry’s

limitations?

E. Janine was measuring the table as on the right.

What is wrong?

What can be done to help Janine?

What other situations like this can occur in measurement where

difficulties other then conceptual ones cause problems? List

come cases! [Remember discipline, psych-motor skills, etc.]



F. Joe could find the area of the left figure because he could use the formula [A = L x B], but he could not

find the area of the right figure. As he said: “I don’t have a formula for that shape!”.

What is missing from Joe’s concept of area?

What experiences would remediate Joe?

G. Sue was puzzled. Yesterday she had poured water from the jar to the measuring cup without changing

its volume? She could not understand why today she could not find the area of her geostrip

parallelogram by simply straightening it up into a rectangle and using the formula A = L x B.

Is there any justification for Sue’s puzzlement?

What would you say to Sue? What would you do with her to show her that she could not do as she wanted?

H. John looked at the illustration on the right and said that the container on the left had the greater mass!

(1) Is John’s answer as you expected?

(2) Is John’s answer necessarily correct?

(3) How could you make John aware of the problem with the question as it is posed? How could the question be

posed to avoid ambiguity?

(4) If, even when all ambiguity is removed, John still believes that larger objects are heavier, what experiences

would you give him to overcome this misconception?



I. Tom sorted the coin, book, chair, scissors, pencil and table into two groups: large objects – table, chair

and book; small objects – scissors, pencil and coin. Mary sorted the same objects into: large – table

and chair; small – scissors, book, pencil and coin.

(1) Which of these two results is correct?

(2) Why does this situation emerge?

(3) What does this say about correct answers in some measurement situations?

(4) What important measurement results can emerge this beginning?

J. Bill was not interested in the sundial project. “Why did they make sundials?” he said. “Why didn’t they

just look at their watches?”

(1) What is lacking in Bill’s measurement experiences?

(2) Is an historical perspective important? Why?

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